Understanding Plane Waves 1536167797, 9781536167795

Table of contents :
Contents
Preface
1 Recent Progress in Acoustical Theory and Applications • Lin Fa and Meishan Zhao
2 Recent Development of an Acoustic Measurement System • Lin Fa and Meishan Zhao
3 Plane Nonlinear Elastic Waves: Approximate Approaches to Analysis of Evolution • Jeremiah Rushchitsky
4 Spacetime Symmetries and Interaction of Quantum Relativistic Particles with External Plane Wave Fields • H. K. Ould-Lahoucine
Index
Related Nova Publications

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PHYSICS RESEARCH AND TECHNOLOGY

UNDERSTANDING PLANE WAVES

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PHYSICS RESEARCH AND TECHNOLOGY Additional books and e-books in this series can be found on Nova’s website under the Series tab.

PHYSICS RESEARCH AND TECHNOLOGY

UNDERSTANDING PLANE WAVES

WILLIAM A. COOPER EDITOR

Copyright © 2020 by Nova Science Publishers, Inc. All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. We have partnered with Copyright Clearance Center to make it easy for you to obtain permissions to reuse content from this publication. Simply navigate to this publication’s page on Nova’s website and locate the “Get Permission” button below the title description. This button is linked directly to the title’s permission page on copyright.com. Alternatively, you can visit copyright.com and search by title, ISBN, or ISSN. For further questions about using the service on copyright.com, please contact: Copyright Clearance Center Phone: +1-(978) 750-8400 Fax: +1-(978) 750-4470 E-mail: [email protected].

NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers’ use of, or reliance upon, this material. Any parts of this book based on government reports are so indicated and copyright is claimed for those parts to the extent applicable to compilations of such works. Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the Publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS. Additional color graphics may be available in the e-book version of this book.

Library of Congress Cataloging-in-Publication Data ISBN: HERRN

Published by Nova Science Publishers, Inc. † New York

CONTENTS Preface Chapter 1

Chapter 2

Chapter 3

Chapter 4

vii Recent Progress in Acoustical Theory and Applications Lin Fa and Meishan Zhao

1

Recent Development of an Acoustic Measurement System Lin Fa and Meishan Zhao

113

Plane Nonlinear Elastic Waves: Approximate Approaches to Analysis of Evolution Jeremiah Rushchitsky

147

Spacetime Symmetries and Interaction of Quantum Relativistic Particles with External Plane Wave Fields H. K. Ould-Lahoucine

203

Index

217

Related Nova Publications

221

PREFACE As a critical theoretical advance, Understanding Plane Waves discusses the acoustic Goos-Hänchen effect. The important applications of this effect are discussed, including plane wave propagating inside transversely isotropic elastic-solids, reflection/refraction at interface between two anisotropic rocks, and acoustical applications to petroleum logging and seismic exploration. Next, the authors explore a newly developed acoustic-measurement system with emphasis on measurement process and recent improvements that make an acoustic-measurement more accurate. Three approaches which are used to analyze the evolution of the plane longitudinal and transverse waves that are propagated in a nonlinear hyperelastic medium are discussed: the method of successive approximations, the method of slowly varying amplitudes and the method of restriction on the displacement gradient. Lastly, the subject of relativistic quantum particles interacting with classical plane wave fields is examined from the standpoint of space-time symmetries which have been found to be encoded in the solutions of relativistic equations. Chapter 1 - Studies of plane wave propagating inside applied media have been central to the development of our understanding of acoustics, so as to the development of applied technologies. It was understood from the

viii

William A. Cooper

beginning of these studies that a detailed description of plane wave propagating inside anisotropic media is extremely important. Recent research progress in this aspect has attracted the attention of scientists from both the physical sciences and the biological sciences divisions. In this chapter, the authors begin with introducing some basic but important concepts of plane wave propagating inside isotropic/anisotropic media, including electric-acoustic conversions, elliptical-polarization states, anisotropy effect on time-depth relation, and more. As a critical theoretical advance, the authors will discuss the acoustic Goos-Hänchen effect. The important applications will be discussed, including plane wave propagating inside transversely isotropic elastic-solids, reflection/refraction at the interface between two anisotropic rocks, and acoustical application to petroleum logging and seismic exploration, e.g., slim-hole acousticlogging tool, conversions of acoustic logging signal, as well as seismic signal and data analysis. The authors end the chapter with some concluding remarks and speculations. Chapter 2 - The authors discuss a newly developed acousticmeasurement system with emphasis on measurement process and recent improvements that make an acoustic-measurement more accurate. This system is based on a model electric-acoustic transmission-network which consists of a series of parallel-connected equivalent-circuits. For acoustictransducers, the authors place special emphasis on the importance of the contribution from each individual frequency component of an excitation signal, the propagation medium, and the cumulative signal-output from transducer’s mechanic/electric terminals. The system has been tested for realistic acoustic transmission. Based on the results of the measurement, the authors conclude that this system is easy to operate, user friendly, and highly accurate. Chapter 3 - Three approaches (methods) are used to analyze the evolution of the plane longitudinal and transverse waves that are propagated in a nonlinear hyperelastic medium - method of successive approximations, method of slowly varying amplitudes, method of restriction on the displacement gradient. The evolution is understood in the standard for physics meaning: the wave with some given initial profile (harmonic or

Preface

ix

solitary) evolves, that is, changes this profile. The term “the profile is distorted” is used sometimes too. The medium of propagation is described by the well-known in nonlinear mechanics of materials five-constant Murnaghan model. First, the noted methods are described briefly as applied to the wave propagation problems. Further, the longitudinal and transverse plane waves are analyzed separately. The point is that the Murnaghan model describes the longitudinal wave by the quadratic nonlinear wave equation, whereas the transverse wave – by the cubic nonlinear wave equation. Ten variants (known and new = published and unpublished) for the longitudinal wave and four variants for the transverse wave (known and new = published and unpublished) of an approximate analysis are described and commented on. It is shown that each variant provides an answer on a certain aspect in the initial wave profile evolution study. A statement of some variants is accompanied by the 2D and 3D pictures. An attention is drawn to the features of the evolution process as well as to similarities and differences in the results obtained. Chapter 4 - The subject of relativistic quantum particles interacting with classical plane wave fields is examined from the standpoint of the space-time symmetries which have been found to be encoded in the solutions of relativistic equations. Principally, it is shown how the elements of the proper Lorentz group come into play as a basic ingredient to get the solution of the Dirac equation under the form of variable transformations acting on the free-field solution. Subsequently, this underlying Lorentz structure is also found in the full solutions of spin 1 particles in interaction with classical plane wave fields by mean of a local gauge, a Lorentz and displacement transformations (ULT) acting as variable transformations on the free-field. On the other hand, considering the role played by the relativistic Green function as a fundamental object in the description of several scattering processes in quantum electrodynamics (QED) involved with the electromagnetic plane waves, an exhaustive review is done of the different approaches devoted to its derivation including the algebraic methods, path integrals and worldline formalisms.

In: Understanding Plane Waves Editor: William A. Cooper

ISBN: 978-1-53616-779-5 © 2020 Nova Science Publishers, Inc.

Chapter 1

RECENT PROGRESS IN ACOUSTICAL THEORY AND APPLICATIONS Lin Fa1 and Meishan Zhao2,* 1

School of Electronic Engineering, Xi’an University of Posts and Telecommunications, Xi’an, Shaanxi, China 2 James Franck Institute and Department of Chemistry, University of Chicago, Chicago, IL, US

ABSTRACT Studies of plane wave propagating inside applied media have been central to the development of our understanding of acoustics, so as to the development of applied technologies. It was understood from the beginning of these studies that a detailed description of plane wave propagating inside anisotropic media is extremely important. Recent research progress in this aspect has attracted the attention of scientists from both the physical sciences and the biological sciences divisions. In this chapter, we begin with introducing some basic but important concepts of plane wave propagating inside isotropic/anisotropic media, *

Corresponding Author’s Email: [email protected].

2

Lin Fa and Meishan Zhao including electric-acoustic conversions, elliptical-polarization states, anisotropy effect on time-depth relation, and more. As a critical theoretical advance, we will discuss the acoustic Goos-Hänchen effect. The important applications will be discussed, including plane wave propagating inside transversely isotropic elastic-solids, reflection/refraction at the interface between two anisotropic rocks, and acoustical application to petroleum logging and seismic exploration, e.g., slim-hole acoustic-logging tool, conversions of acoustic logging signal, as well as seismic signal and data analysis. We end the chapter with some concluding remarks and speculations.

Keywords: acoustical measurements, measurement network, anisotropic media and interface, reflection and refraction, polarization states, anomalous incident angle, Acoustic Goos-Hänchen effect, acousticallogging, seismic signal and analysis

1. INTRODUCTION Developing acoustic technologies for successful application requires a comprehensive understanding of acoustic theories and sometimes significant extensions of the conventional approaches. A rigorous extension of the conventional approaches to reduce or eliminate errors in practical applications is challenging. It requires factual identification and proper ranking of the influential factors involved in theory and in practical world. For example, identifying the influences of physical and geometric factors on reflection/refraction coefficients in systems of interface is far from trivial. There are many competing factors involved, from transition media to measurement system to the transmitting waves. Acoustic wave bears many similarities to optical or electromagnetic wave in propagation, reflection, refraction, and polarization. For example, linear conversion and anomalous refraction for electromagnetic waves was reported by Grady et al. [1]; the anomalous reflection/refraction of light and its propagation with phase discontinuities was reported by Yu et al. [2]; Fa et al. predicted the existence of an anomalous incident-angle for an acoustic, inhomogeneously refracted P-wave [3] at solid-solid interface.

Recent Progress in Acoustical Theory and Applications

3

Obviously, a successful acoustic theory embraces a good understanding of the similarities and the differences in important features to optics, e.g., the Goos-Hänchen effect. This brings up imperative questions. For example, what are the similarities of acoustic wave propagating onto a fluid–solid interface comparing to optical waves? Is there truly an acoustical GoosHänchen effect? If yes, can we provide a theoretical base and/or positive evidence from practical applications? How does an anomalous incidentangle affect the nature of an acoustic wave at a solid-solid interface? By answering these questions, we wish to achieve an improved understanding of acoustics. In geophysical applications, the interior of the earth is commonly modeled as a series of isotropic, regular thin layers with distinctive physical properties, e.g., a sequence of strata with a vertical-axis symmetry (VTI model), a sequence of tilt rock-layers without a vertical-axis symmetry (TTI model), etc. In these media, the macroscopic anisotropy and mechanical property are usually described by elastic tensors of hexagonal system crystals. The matter is that macroscopic anisotropy may provide a significant influence on propagation, reflection/refraction, and polarization of the seismic waves. Indeed, in studies of solid-solid interfaces, many influential factors on reflection/refraction coefficients have been positively identified, e.g., incident-angle of wave, physical, anisotropic parameters of media, etc. In turn, it leads to significant improvement in scientific and industrial applications, such as amplitude variations with offset (AVO) analysis of seismic data in petroleum industry, wave energy device development, rangefinder, medical imaging, underground geological structure imaging, nondestructive detection of dam and high-rise building foundation, underwater acoustics detection, and more. Essentially, these studies provided the foundation for seismic forward modeling and seismic amplitude variations with offset data analysis. Studies and applications to the field of geoscience have been reported extensively. Thomson [4] and Wang [5, 6] reported measured rock anisotropy and modeled rock anisotropy by a stiffness matrix of a hexagonal system crystal. Reflection coefficients from VTI-VTI interface

4

Lin Fa and Meishan Zhao

have been reported by several groups, e.g., Daley et al. [7], Rüger [8], Tsvankin [9], Carcione [10], and Klimeš [11]. Higher-order polynomials in analysis of reflection/refraction angle induced at interfaces of various media have also been developed by researchers [12-14]. Nevertheless, these results preclude the influences of physical and geometric parameters of a measurement system, nor do they provide explicit information of elliptical-polarization states. Therefore, much work in this direction remains to be done. A key element is a good measurement system and its associated physical and geometrical parameters. So, technically, for experimental measurement, we need an intelligent choice of these parameters. For example, do we need a spherical shell transducer, a cylindrical transducer, a flake transducer, or other kinds of transducers? What are the effects of these parameters on the measured acoustical signal? All these are important considerations in practical applications. This chapter, mostly focused on the work from our group, is organized as follows. We begin with discussions of theoretical development on the Goos-Hänchen effect in acoustics, through an acoustic wave scattering on a fluid–solid interface. Numerical calculations and analysis have been performed based on a modeled water-Perspex interface. Specifically, we discuss the reflection/refraction coefficients and the first arrival of the acoustic wave, and then find lateral displacement and transition time. In Section III we discuss the existence of an anomalous incident-angle at solid-solid interfaces and related issues on elliptically rotational polarization. In Section IV, we focus on elliptical-polarization states of an elastically refracted P-wave, arising from a P-wave impinging on an interface between two VTI media with strong anisotropy, which provide us an enhanced understanding of acoustical-wave scattering. In Section V we discuss applications to acoustic-logging and related issues. Finally, Section VI contains an added discussion of the solid-solid interface, including the VTI-TTI interface. We provide, in Section VII, some concluding remarks and speculations.

Recent Progress in Acoustical Theory and Applications

5

2. ACOUSTIC GOOS-HÄNCHEN EFFECT 2.1. Acoustic Wave Scattering on Fluid–Solid Interface The physical mechanism of Goos-Hänchen effect in optics [15] is generally understood as follows that for an optical wave propagating from an optically denser medium to an optically thinner medium, the total reflection generates a coherent interference and synthesizes a propagating wave with a lateral displacement with respect to the incidence location on the interface. This phenomenon plays an important role in optical interfacial transition. Applications of Goos-Hänchen effect in optics are readily available in the literature [16–25]. Attempted extensions of Goos-Hänchen effect to acoustics have also been published elsewhere [26-31], e.g., nondestructive detection, seismic exploration, and acoustic logging. In this section, we discuss models of lateral displacement of acousticwave scattering on fluid-solid interfaces and present an analog of the Goos-Hänchen effect in acoustics. Consider a harmonic P-wave impinges on a fluid–solid interface, as shown in Figure 1 (m=0). This action yields several reflected/refracted waves, i.e., reflected P-wave, refracted P-wave, and refracted SV-wave [3], which are denoted by an index m ={1, 2, 3}.  ( m) is a corresponding geometric angle and R ( m ) is a reflection/refraction coefficient. Assume that an acoustic source radiates an acoustic wave omnidirectionally, then a Ricker wavelet may be used as such a wave which contains different frequency components, as shown in Figure 2. We select the parameter h0  16 104 with a central frequency of 1.0 MHz. A Ricker wavelet [32] is given in time and frequency domains as 2 s(t )  1  2 2h02t 2  exp   h0t    

S ( ) 

 2   2 exp  2 2  2 3h03  4 h0 

(1)

(2)

6

Lin Fa and Meishan Zhao

Figure 1. Reflection and refraction of a harmonic acoustical wave at fluid–solid interface.

Figure 2. Waveform and amplitude spectrum of a Ricker wavelet: (a) waveform; (b) amplitude spectrum.

2.2. Reflection/Refraction To simplify the discussion, we analyze the interface with a critical incident angle c(2) corresponding to the refracted P-wave. We are interested in the reflection coefficients from a fluid–solid interface with at least one critical incident angle. As an example, let’s consider water– Perspex interface. The physical parameters of this interface are presented in Table 1. In this system, there is one known critical incidence angle, located at c(2)  31.83 .

Recent Progress in Acoustical Theory and Applications

7

For incidences in the post-critical-angle region, a reflected harmonic Pwave will generate a phase shift with respect to the incidence P-wave, which is given by

   arctan

a1 (d1  b1 ) a12  b1d1

(3)

where a1  

b1 

1 v v

(2) (3)

 sin

2

c11( s ) cos2  (2)  c12( s ) sin 2  (2)  cos 2 (3)

 (2)  1

1/ 2

(s)  2c44 (2) (3)  (2) (3) sin  sin 2 v v 

  c( f )  2c ( f )   (1)11 (3) cos 2 (3)  (1) 11(2) sin  (2) sin  (3)  cos  (1)  v v  v v 

and d1 

 sin

2

 (2)  1

1/ 2

(s)  2c44 (2) (3)  (2) (3) sin  sin 2 v v

  c( f )  2c ( f )   (1)11 (3) cos 2 (3)  (1) 11 (2) sin  (2) sin  (3)  cos  (1)  v v v v   2 

Table 1. Physical parameters in water–Perspex interface Perspex

Water

P-wave propagation velocities (m/s)

2709.0

1428.6

SV-wave propagation velocities (m/s)

1365.0

density of Perspex (g/cm3)

1.17

1.00

8

Lin Fa and Meishan Zhao

2.3. Lateral Displacement and Transition Time Let’s discuss two models, e.g., the virtual lateral displacement model (VLDM) and the real lateral displacement model (RLDM), shown in Figure 3. VLDM (dashed line) has a propagation path T  A  RI and RLDM (solid line) has a propagation path T  A  B  RR [see Figure 3(b)]. All frequency components in VLDM have the same propagation path but each frequency component has its own unique transition time. For RLDM as shown in Figure 3(c), different frequency components have both different propagation paths and different transition time. In the following, we present a detailed analysis by using the results calculated from VLDM. The discussion and analysis of RLDM is similar.

(a)

(b)

(c)

Figure 3. (a). Traditional sliding-refraction P-wave acoustic-logging model, where T is the acoustic source, R is the observation position; (b). Schematic presentation of VLDM (dashed line) and RLDM (solid line); (c). Propagation paths of acoustic signal wavelet for the RLDM.

Recent Progress in Acoustical Theory and Applications

9

Presented in Figure 3(b), the incidence point (A) differs from the reflection point (B) on the interface with a virtual/real lateral displacement AB . From equation (3), the lateral displacement of the reflected harmonic

P-wave can be written as [30, 31]

z  (0) , f   

a1 b d  a1  a12  d12  1  a1  a12  b12  1 k z k z k z (4) 2 2 2 2  a1  b1d1   a1  d1  b1 

(d1  b1 )  b1d1  a12 

where z( (0) , f )   / kz , a1 / kz  c1 / kz , and 2   v(2) v(3)  a1 4  ( s ) (2) 2 ( s ) (3) 2  sin  (0) ( s)  3 (0)   sin    c44  v   c11  v   (0)  4c44 (0) 3 k z v(2) v(3)  v v    

b1  k z

4c44( s ) v(2) v(3)

 2v(2) v(3) v(2) (v(3) )3 sin 3  (0)  (0) 2 (3)  (0) sin  1  sin    (v(0) )3 1  sin 2  (3)   v

 sin 2  (2)  1 v(3) sin 3  (2)

1  sin 2  (3)   sin 2  (2)  1 



 2(v(3) )2 sin 3  (0) 2(v(3) ) 2  c  sin 2 (0)  sin 2  (2)  1  (0) (0) (0) v v   v cos  v 



(v (2) v (3) )2 sin 2 (0) sin 3  (0) (v (0) )3 sin 2  (2)  1

(f) 11 (0) (3)

v(0)

sin  (0) (v(2) )2 sin 2  (2)  1  (0) cos  2v(0)

  1 

sin 2 (0) sin  2

(2)

10

Lin Fa and Meishan Zhao 2c11( f ) v (3)  sin  (0)  3sin 2  (0)  2  sin 2  (2)  1  v (0)  cos  (0)

 v (2)  cos  (0) sin 3  (0)     (0)  sin 2  (2)  1  v  2

d1  k z

(s) 4c44 (2) (3) v v

 2v (2) v (3) sin  (0) 1  sin 2  (3)  (0) v  sin 3  (0)

 2 (2)  sin   1 



v (2) (v (3) ) 2 (v (0) )3



(v(2) )3 v(3) 3 1  sin 2  (3)  sin   (v(0) )3 sin 2  (2)  1 



 c11( s )  2(v (3) ) 2  sin 3  (0)  sin 2 (0)  sin 2  (2)  1  (0) (3)  (0) (0) v v  v  cos  



(3)

(v (2) v (3) ) 2 sin 2 (0) sin 2  (0) (v (0) )3 sin 2  (2)  1

v (0)



1  sin  2

sin  (0) cos  (0)

sin 2  (2)  1 

(v (2) ) 2 2v (0)

  1 

sin 2 (0) sin  2

(2)

2c11( f ) v (3)  sin  (0) (3sin 2  (0)  2) sin 2  (2)  1  (v (0) ) 2  cos  (0)

Recent Progress in Acoustical Theory and Applications

11

1  v(2)  sin 2 (0) sin 2  (0)     (0)  2v  sin 2  (2)  1  2

The transition time is t ( (0) , f )   (1) /  which yields

t  , f    (0)

a1 b d  a1b1  a12  d12  1  a1  a12  b12  1    2 2 2 2 a  b d  a d  b  1 1 1 1  1 1

(d1  b1 )  b1d1  a12 

where   (2) (3) (3) a1 4sin  (0)  ( s ) v(2) (s) v ( s) v v 2 (0)   c  c  4 c sin  44 11 44 2 2   v(3) v(2)   v(0)    v(0)  

4c44( s ) sin  (2) sin  (3)  sin 2  (2)  1 b1   v(2) v(3)

1/2

 sin 2  (3)   2cos  (3)  cos  (3) 

 sin 2  (2) cos  (3)    2 (2)   sin   1 

2c11( f ) sin  (2) sin  (3)  sin 2  (2)  1

1/ 2



v(0) v(2)

 sin 2  (0) cos  (0) sin 2  (2)  (0)      2 cos    (0) 2 (2)  cos    sin   1  2c11( f ) sin  (2) sin  (3)  sin 2  (2)  1

1/ 2



v(0) v(2)

(5)

12

Lin Fa and Meishan Zhao  sin 2  (0) cos  (0) sin 2  (2)  (0)      2 cos    (0) 2 (2)  cos    sin   1 

and d1 4c ( s ) sin  (2) sin  (3)  sin 2  (3)  (3)   44 sin 2  (2) -1  2 cos  (2) (3) (3)   v v cos   



sin 2  (2) cos (3)   sin 2  (2) -1 



 c11( f ) sin 2  (2)  1   sin 3  (0) (2sin 2  (3)  1)  4cos  (0) sin 2  (3)   (0) (3) (0) v v   cos  



cos  (0) sin 2  (2) (2sin 2  (3)  1)    2 (2)  sin   1 

 2c11( f ) sin  (2) sin  (3)  sin 2  (0)  2 cos  (0)  sin 2  (2)  1  (0) (2) (0) v v   cos 



cos  (0) sin 2  (2)   sin 2  (2)  1 

where k z  k z( 0 )  k z(1) . Both the lateral displacement and the transition time of a reflected harmonic P-wave are inversely proportional to its frequency. Meanwhile, the effective propagation speed of a reflected harmonic wave along the fluid–solid interface can be calculated as the ratio of lateral displacement to the transition time

Recent Progress in Acoustical Theory and Applications v  (0)  

z  (0) , f  t  (0) , f 

13 (6)

Figure 4 shows the calculated lateral displacement (  z ), transition time ( t ), and propagation speed of a reflected harmonic P-wave at the water-Perspex interface. The spectra of lateral displacement and transition time are following a similar trend as a function of the incident angle. Both have a local maximum at  (0) =37.58 .

Figure 4. Lateral displacement, transition time, and propagation speed of a harmonic P-wave reflected from the water–Perspex interface: (a) lateral-displacement, (b) transition time, (c) propagation speed of the harmonic reflected P-wave along the water–Perspex interface.

The line-segments TM1 , M 1 M 2 and M 2 R in Figure 3(a) is the propagation path of the so-called sliding refraction P-wave in the traditional acoustic-logging model. Figure 3(c) shows the propagation path

14

Lin Fa and Meishan Zhao

of each frequency component in the acoustic signal wavelet for RLDM. The lateral displacement and transition time are positive, which means that the time needed going through the paths in Perspex is longer than the needed time along the propagation path inside water. This is a concrete evidence that there is a virtual/real lateral displacement induced in the process of acoustic wave impinging on the water–Perspex interface, i.e., an acoustic Goos–Hänchen effect at the fluid–solid interface.

2.4. First Arrival of the Acoustic Signal The sliding refraction P-wave model has been used traditionally for acoustic-logging in geophysical applications [33–40]. This model has been popular and been used to calculate the propagation speed of a P-wave in the formation around a borehole in acoustic logging. Following propagation path of a refracted sliding P-wave, the solid-line in Figure 3(a), its transmission time can be written as ts 

2a 2a 2a  tan  ( 2 ) tanc( 2 ) v( 0 )cosc( 2 ) v( 2 ) v

(7)

For fluid–solid interface with a virtual lateral-displacement propagation T  A  RI , each frequency component of the acoustic wave has its own unique transition time. At a given observation point RI , the jth frequency component can be written as  y j (t )  Re | S ( j ) || R (1) ( (0) ) | 

  cos  (1)    2ak j   exp i   j t   j t j  (1) cos  (0)    sin  

 2ak j  H  t  t j     j cos (0)  

      

Recent Progress in Acoustical Theory and Applications

15

2ak j   2ak j   S ( j ) R(1)  (0)  cos  j  t  t j   H  t  t j  (0)  cos    j cos (0) 

  

(8) where | S ( j ) | , t (lj ) , and k j are the corresponding amplitude, transition time, and wave number. H(·)is a Heaviside unit step-function. The acoustic signal wavelet is a summary of all frequency components N

yI (t )   y j (t )

(9)

j 1

The geometrical structure of a measurement system is denoted by an angle parameter ξ. As shown in Figure 3(b), along the propagation path T  A  B  RR , different frequency components of the acoustic signal propagate along different paths. The jth frequency component follows the path T  O j  O  O'j  RR [see Figure 3(c)]. All propagation paths satisfy a unique geometric constraint, which guarantees a positive determination of the propagation paths, F  (0) , f   z  (0) , f   2a tan  (0)  L  0

(10)

where L is the shortest distance from the radiation source to the observation point; a is the shortest distance from the observation point to the fluid–solid interface; and z( (0) , f ) is the lateral displacement of a frequency component. It is noted that for some frequency components with a given angle parameter ξ, there could be more than one solution from equation (8) along a propagation path. At the observation point RR, the jth frequency component of the acoustic signal along the sth propagation path can be expressed by

16

Lin Fa and Meishan Zhao   y j .s (t )  Re  S ( j ) R (1)  (0) j,s   

  cos  (0)    2ak j j,s   exp i   j (t  t j , s )  (0)   sin   cos  (0)   j,s j,s  

 2ak j H  t  t j , s    j cos  j(0),s 

    

  

 2ak j   S ( j ) R (1)  j(0)  , s  cos  j  t  t j , s   cos  (0)  j    2ak j H  t  t j , s  (0)   j cos  j 

  

(11)

where M is the number of propagation paths of each frequency component; (l ) th  (0) j , s and t j , s are the incident angle and transition time of the j frequency

component. At the observation point RR, the reflected acoustic wave is a summation of all frequency components N

M

yR (t )   y j.s (t ) j 1 s 1

(12)

The extra transition time of the acoustic wave reflected from waterPerspex interface may be viewed as an alternative measurement for lateral displacement. For a VLDM path T  A  RI [see Figure 3(b)], we define the acoustic-signal transition time as tI  tI  t p , where t I is the calculated acoustic-signal transmission time from T to A, plus the time from A to RI, where t p is the pure propagation time of the acoustic wave in water along the same path. The deviation of the transmission time from the traditional sliding refraction P-wave model can be calculated from tIs  tI  ts . Figure 5 shows that there is indeed a virtual/real lateral displacement induced by the acoustic wave impinging on water-Perspex interface, i.e., again a solid evidence of acoustic Goos-Hänchen effect at the fluid–solid interface.

Recent Progress in Acoustical Theory and Applications

17

Figure 5. Waveform, transition time, and transmission time-deviation from the traditional sliding refraction P-wave model, where θ(0) = ξ: (a) waveform yI at observation point RI , with θ(0) = 40°, (b) transition time  t I , (c) transmission time deviation t Is .

3. ANOMALOUS INCIDENT-ANGLE 3.1. VTI Media with Strong Anisotropy A macroscopic transversely isotropic medium with a vertical axis of symmetry is named as a VTI medium. This structure is generically concerning the interior of the earth, where the sequence of the layers of the earth behaves anisotropic, whereas still transversely isotropic [4, 10, 41, 42]. Studies of reflection of acoustic wave with respect to such media is important in geophysics, as well as in seismic image and analysis [43-47].

18

Lin Fa and Meishan Zhao

The physical property of a VTI medium is usually described by an elastic stiffness tensor of a hexagonal crystal [10, 42, 45].

CVTI

 c11 c12 c13 c  12 c11 c13 c c c   13 13 33 0 0 0 0 0 0   0 0 0

0 0 0 c44 0 0

0 0 0 0 c44 0

0 0  0  0 0  c66 

(13)

where c12  c11  2c66 , cij is a modulus with respect to stress and strain in the medium, and (i, j) = (1, 2, 3, 4, 5, 6). In terms of anisotropic rock parameters, the matrix elements of the stiffness tensor are given by [4]



2 2 2 2 c13( n )   ( n )  ( n )     ( n )    ( n )  1 ( n )     ( n )      

  *( n )  ( n )  c33( n )   ( n )  ( n ) 



4 1/2

  (n)   (n) 

2

2

c11( n)   2 ( n)  1  ( n)  ( n)  (n) c44   ( n)   ( n) 

2

2

c66( n )   2 ( n )  1  ( n)  ( n) In absence of body forces, the Christoffel equation can be written as [9, 10, 48]

Recent Progress in Acoustical Theory and Applications ( m) ( m)  11  u x( m)   [v( m ) ]2 0 13   ( m)  ( m) 0 (22m )  [v( m ) ]2 0   u y  R  0  ( m) ( m) ( m ) 2  ( m)   31 0 33  [v ]    u z 

19

(14)

where the subscript {n}={1, 2} stands for incidence medium or refraction medium and the superscript {m}={0, 1, 2, 3, 4} denotes incident P-wave, reflected P-wave, refracted P-wave, reflected SV-wave, and refracted SVwave, respectively; and ( m) ( m) 13  31

( m) 13  [ A13( n)  A44( n) ]sin  ( m) cos ( m)

( m) 11  A11( n) sin2  ( m)  A44( n) cos2  ( m)

( m) 33  A33( n) cos2  ( m)  A44( n) sin 2  ( m)

A(jin )  C (jln ) /  ( n ) Solving Christoffel equation leads to the uncoupled dispersion relations  ( m )   v ( m ) 2    ( m )   v ( m ) 2   0  11   33 

(15)

(22m)   v( m)   0

(16)

2

then, yields the phase velocities of incident P-wave and the induced waves v1(,m2 )  {[ A4( m ) sin 2  ( m )  A5( m )  Q (m) ( ( m ) )] / 2}1/ 2 (17) Q ( m) ( (m) )  [( A1( m) sin 2  ( m)  A2(m) cos2  ( m) ) 2  ( A3( m) ) 2 sin 2 2 ( m) ]1/ 2 (18)

20

Lin Fa and Meishan Zhao v3( m )    (22m ) 

1/ 2

(19)

In equations (15)-(19),  (n ) is the density of incidence and refraction media; the subscripts {j, l}={1,2, 3, 4, 5, 6}, the subscripts 1, 2, and 3 indicate three different solutions of the phase velocity ( v1(,m2,)3 ); also ( n) ( n) A1( n)  A11  A44

( n) ( n) ( n) ( n) ( n) ( n) , A2(n)  A44 , A3(n)  A13 , A4(n)  A11 ,  A33  A44  A33

( n) ( n) , and A (jln )  C (jln ) /  ( n ) . A5( n)  A33  A44

For a P-wave propagating in xz-plane, we look at the interface between  two VTI media in Figure 6, where S (m ) are displacements of incident Pwave and the induced waves at the interface between two VTI media. Table 2. Anisotropic parameters for A-shale and O-shale rocks, where A-shale is the incidence medium and O-shale is the refraction medium Medium

α (m/s)

A-shale 2745

O-shale 4231

β(m/s)

1508

2539

ρ(g/cm3)

2.340

2.370

ε

0.103

0.200

δ*

-0.073

0.000

γ

0.345

0.145

21.264

9.397

6.976

15.824

17.632

42.426

5.321

15.278

8.993

19.709

Thomsen Parameters:

Elastic constants (GPa):

c11 c13 c33 c44 c66

Recent Progress in Acoustical Theory and Applications

21

Figure 6. Schematic representations of polarization vectors and wave-front normal for incident P-wave and induced waves at the interface. The long rays indicate the phase velocity direction and the short rays show the polarization direction.

Now, let’s consider the interface between an anisotropic shale (Ashale) and oil shale (O-shale). Table 2 contains the physical parameters of these sedimentary rocks [4, 6, 13, 48]. A P-wave starting from A-shale propagates to an interface with O-shale which is the refraction medium. (m) For a harmonic acoustic-field, the wave displacements S can be written as

S

(0)

 u x(0)   R  (0)  exp[i (t  k (0) x sin   k (0) z cos  )]  u   z  (0)

 uxo(1)  S (1)  R (1)  (1)  exp[i (t  k (1) x sin  (1)  k (1) z cos  (1) )] u   zo   u x(1)  | R |  (1)  exp[i(t  k (1) x sin  (1)  k (1) z cos  (1) )] u   z  (1)

(20)

(21)

22

Lin Fa and Meishan Zhao

S

(2)

 uxo(2)   R  (2)  exp[i(t  k (2) x sin  (2)  k (2) z cos  (2) )]  u   zo  (2)

 ux(2)  | R |  (2)  exp[i(t  k (2) x sin  (2)  k (2) z cos  (2) )]  u   z 

(22)

(2)

 uzo(3)  S (3)  R (3)  (3)  exp[i(t  k (3) x sin  (3)  k (3) z cos  (3) )]  u   xo   u z(3)  | R |  (3)  exp[i(t  k (3) x sin  (3)  k (3) z cos  (3) )]  u   x 

(23)

(3)

S

(4)

 uzo(4)   R  (4)  exp[i(t  k (4) x sin  (4)  k (4) z cos  (4) ) u   xo  (4)

 uz(4)  | R |  (4)  exp[i(t  k (4) x sin  (4)  k (4) z cos  (4) )] u   x 

(24)

(4)

and  ux( m )   uxo( m )   uxo( m) (cos  ( m)  i sin  ( m) )   a1( m)  ib1( m)  (m)   ( m)  ( m )    ( m )  exp[i ]   ( m) ( m) ( m)  ( m)   uzo (cos   i sin  )   a3  ib3  (25)  uz   uzo 

where the superscripts {m}={0, 1, 2, 3, 4} denote the incident P-wave (m = 0), reflected P-wave (m = 1), refracted P-wave (m = 2), reflected SVwave (m=3), and refracted SV-wave (m=4), respectively;  (m ) is the (m) incidence/reflection/refraction angle for the corresponding wave; u xo and ( m) are the modulus of polarization coefficients of each wave; R(1) is the uzo

reflection coefficient of P-wave to P-wave; R(2)is the refraction coefficient of P-wave to P-wave; R(3)is the reflection coefficient of P-wave to SVwave, R(4) is the refraction coefficient of P-wave to SV-wave;  (m) is a phase

Recent Progress in Acoustical Theory and Applications shift ( m)

R

for ( m)

=| R

an

induced wave | exp[i ( m) ] and R(0)=1.

relative

to

the

incident

23 P-wave;

The critical incident-angle is denoted by c(2) and an anomalous incident-angle is denoted as a(2) which is in the post critical incidence region (c(2) , 90 ) .

3.1. An Anomalous Incident-Angle In this section, we are going to examine the system for anomalous incident angle to see if it indeed exists. If it does exist, then to find it. For an incident-angle in the region before a critical incident angle, (0)   c(2) , the reflection/refraction coefficients are real (not complex). In the range of  (0)  c(2) , the reflection/refraction coefficients are complex (not real). Based on the Christoffel equation, the phase velocities for incident P-wave and the induced waves are given in equation (17). In post critical anomalous incident-angle a(2) region (c(2) , 90 ) , the phase velocities of the refracted P-wave satisfy Snell’s law, sin  (2) / vi(2)  sin  (0) / vi(0)

where subscript i = 1 is for  (0) [0 , a(2) ) and i = 2 is for

 (0)  (a(2) , 90 ] . The reflection/refraction angles can be calculated from the forth-order polynomials [12, 13] B1(in) ( (0) )sin 4  (1,3)  B3(in) ( (0) )sin 2  (1,3)  B5(in)  0

(26)

B1( re) ( (0) )sin 4  (2,4)  B3( re) ( (0) )sin 2  (2,4)  B5( re)  0

(27)

where,

24

Lin Fa and Meishan Zhao

Figure 7. The angle of reflected P-wave versus incident angle: there is an obvious abnormality at 62.04o.

Figure 8. The velocity of reflected P-wave versus incident angle: there is an obvious abnormality at 62.04o.

Recent Progress in Acoustical Theory and Applications

25

B1(in) ( (0) )  A2(in)  [ K1 ( (0) )]2 , B3(in) ( (0) )  2[ A1(in)  A4(in) K1 ( (0) )] B5(in)  A3(in)  [ A4(in) ]2 , B1( re) ( (0) )  A2( re)  [ K2 ( (0) )]2

B3(re) ( (0) )  2[ A1( re)  A4( re) K2 ( (0) )] , B5( re)  A3( re)  [ A4( re) ]2 K1 ( (0) )  2 / const( (0) )  A4(in) , K2 ( (0) )  2 / const( (0) )  A4( re)

const( (0) )  [sin  (0) / v1(0) ]2 If there are anomalous incident angles, they must be part of the solutions from equation (17). For the interface of A-shale with O-shale, there is indeed an anomalous incident-angle found at a(2)  62.04o , which is shown in Figure 7. Figure 8 shows the velocity of the refracted P-wave vs. incident-angle. It again reveals a profound anomalous at a(2)  62.04o .

3.2. Elliptically-Polarized Rotational Direction Change For incident P-wave and the induced waves propagating in xz-plane (see Figure 6), to obtain the polarization coefficients, it solves a simplified Christoffel equation ( m) ( m) 2 ( m)  11  ux( m)   [v1,2 ] 13   0 ( m) ( m) ( m) 2 31 33  [v1,2 ]  uz( m)  

(28)

that gives the relations for the polarization coefficients ( m) 2 11( m )  (v1,2 ) uz( m )  ( m) ( m) ux 13

(29)

26

Lin Fa and Meishan Zhao ( m) ( m) 2 33  (v1,2 ) ux( m )   ( m) ( m) uz 13

(30)

Under the normalization condition

ux( m) [ux( m) ]*  uz( m) [uz( m) ]*  1 we obtain the polarization coefficients for incident P-wave and the induced homogenous waves at the interface, ( m) 2  33( m )   v1,2     ( m) 2  11( m )  33( m )  2  v1,2   

1/ 2

(m) x

1/ 2

u z( m )

2   11( m )   v ( m )    (m) ( m) 2  11( m )  33   2 v    

u

(31)

(32)

Abnormality in the Poynting vector can also be used as an alternative in detecting anomalous incident-angles. The z-component of the Poynting vector can be obtained from the reflection/refraction coefficients [10]. The Poynting vector of an incident P-wave and the induced waves propagating in xz-plane can be written as [5] 1 P( m)   [V ( m) ]*  T ( m) 2

where the particle displacement velocity is given by [V ( m ) ]  Vx( m ) 0 Vz( m ) 

and the stress tensors of the incident P-wave are

(33)

Recent Progress in Acoustical Theory and Applications

T (m)

27

 T1( m ) 0 T5( m )      0 T2( m ) 0   ( m ) ( m)    T5 0 T3 

The z-components of Poynting vectors for the incident P-wave and the induced waves are Pz(0)  



2  k  ( in ) (0) 2 C44  ux   C33( in )  uz(0)   cos (0)

2 



  C44( in )  C13( in )  u x( 0 ) u z( 0 ) sin  ( 0 ) 

Pz(1) 

(34)



2  k (1) | R (1) |2  ( in ) (1) 2 C44  ux   C33( in )  uz(1)   cos  (1)   2

  C44( in ) u x(1) u z(1)  C13( in ) u x(1) u z(1)  sin  (1) 

Pz( 2)  

 k ( 2) R ( 2)

2

 C

( re )

44

2

(35)

| ux( 2) |2 C33( re ) | uz( 2) |2  cos  ( 2)

* *  C44( re )  ux(2)  uz(2)  C13( re ) ux(2) uz(2)   sin  (2)  

Pz  (3)

 k (3) R (3)

2

C

( in )

33

2

ux(3)  C44( in ) uz(3) 2

2



 cos

(3)

  C13 u x(3) u z(3)  C44( in ) u x(3) u z(3)  sin  (3)  ( in )

Pz

( 4)



 k ( 4) R ( 4) 2

2

 C

( re )

33

(36)

(37)

| ux( 4) |2 C44( re ) | uz( 4) |2  cos  ( 4)

  C44( re ) u x( 4 ) (u z( 4 ) )*  C13( re ) (u x( 4 ) )* u z( 4 )  sin  ( 4 ) 

(38)

Detailed examinations of the normalized z-component with respect to the incident P-wave and the induced waves provide us a better understanding of anomalous incident-angles

28

Lin Fa and Meishan Zhao 4

Pz( sum) / max(Pz(0) )   Re( Pz( m) ) / max( Pz(0) )

(39)

m 1

For a refracted inhomogeneous P-wave, the z-component of the polarization has a lag of 90o with respect to its x-component, which is defined as left-rotational elliptical-polarized wave. Otherwise, it is a rightrotational elliptical-polarized wave. The normalized z-component of the Poynting vector vs. incident-angle is presented in Figure 9. It shows that refracted P-wave is a linear leftrotational elliptical-polarized wave before reaching an anomalous incident angle at 62.04o. At that point, it switches to a right-rotational ellipticalpolarized wave. This elliptically-polarized rotational direction change is a direct consequence of the anomalous incident-angle at a(2) =62.04o.

Figure 9. The z-components of Poynting vectors versus incident angle: (a) the zcomponent of incident P-wave Poynting vector, which is real; (b) the summation of real parts of all induced waves’ z-components and the incident P-wave z-component are plotted dashed- and solid-lines. Solid-line and dashed-line stack together before reaching an anomalous incident angle at 62.04o.

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29

4. ELLIPTICAL-POLARIZATION STATE 4.1. Polarization Inside Elastic Hexagonal Solids As discussed in the last section, elastic anisotropy is common in the Earth’s interior [4-6, 9, 10, 42, 33]. Sedimentary rocks, such as shale, are commonly treated as being transversely isotropic with a vertical axis of symmetry (VTI), which may be modeled as a system of hexagonal crystal solids. Broadly speaking, without losing generality, we may call both the VTI media and the hexagonal crystals as hexagonal system solids. In this section we are going to discuss the polarization states inside a system of elastic hexagonal solids, especially the polarization coefficients of a quasi P-wave.

Figure 10. A schematic presentation on xz-plane the wave vector, Poynting vector, and polarization vector.

Now, let’s consider a plane wave propagating in an xz-plane, shown in Figure 10. We define a phase angle  as the angle between wave vector and z-axis, energy (ray) angle  as the angle between Poynting vector and z-axis, and polarization angle  as the angle between the polarization vector and z-axis. The normalized particle displacements of P-wave and SV-wave define a polarization vector. Its x- and z-components are termed as polarization coefficients. For a hexagonal system solid, its elastic

30

Lin Fa and Meishan Zhao

moduli tensor has been presented in equation (13) and discussed in the last section. The polarization angle is obtained as 1/ 2

   v2  u    acrtan  x   acrtan  33 2   uz   11  v 

(40)

4.1.1. Polarization Coefficients Each VTI medium has a certain degree of anisotropy. Therefore, within such a medium, the wave-front is not spherical. The polarization direction of P-wave is perpendicular to that of SV-wave at a given space position. Now, we are going to look in detail the polarization coefficients of P-wave. The Poynting vector of a P-wave propagating on an xz-plane can be written as * * P  V * T 1 V T V T  P x    x* 1 z* 5  2 2  Vx T5  Vz T3   Pz 

(41)

In which, V is the particle displacement velocity and T is a stress tensor. The x- and z-components of Poynting vector of a P-wave can be obtained from equation (41) Px 

k ux  C11ux sin  C13uz cos   u z C44  u x cos  u z sin  2 

Pz 

k ux C44  u x cos  u z sin   u z  C13u x sin  C33u z cos  2 

(42)

(43)

In Table 3, α and β are the propagation speed of P- and SV-waves in direction parallel to the symmetric axis of VTI medium, respectively. Then, the energy angle can be obtained as a function of and polarization coefficients and phase angle

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  atan  Px / Pz   u  C u sin  C13u z cos   u z C44  u x cos  u z sin    atan  x 11 x   ux C44  ux cos  uz sin   uz  C13u x sin  C33u z cos  

(44)

The energy velocity, with respect to phase velocity, is given by [48] ve  v / cos(   )

(45)

Table 3. Physical and anisotropic parameters of rocks and hexagonal crystals Sample

α(m/s)

β(m/s)

ε

δ* ρ(g/m3)

Taylor sandstone (T-Sandstone) Mesaverde mudshale (M-Mudshale) Mesaverde immature sandstone (MISandstone) Mesaverde calcareous sandstone (MCSandstone) Quartz crystal (Q-Crystal) Mesaverde shale (M-Shale) Mesaverde sandstone (3805) (MSandstone-1 Mesaverde sandstone (1582) (MSandstone-2) Mesaverde sandstone (3512) (MSandstone-3) Muscovite crystal (M-Crystal) Biotite crystal (B-Crystal)

3368 4529 4476

1829 2703 2814

0.110 -0.127 2.500 0.034 0.250 2.520 0.097 0.051 2.500

5460

3219

0.000 -0.345 2.690

6096 3383 3962

4481 2438 2926

-0.096 0.169 2.650 0.065 -0.003 2.350 0.055 -0.066 2.870

3688

2774

0.081 0.010 2.730

4633

3231

-0.026 -0.004 2.710

4420 4054

2091 1341

1.120 -1.230 2.790 1.222 -1.437 3.050

The deviations of the polarization angle from the wave vector and the Poynting vector may be used continently in description of the level of polarization     

(46)

32

Lin Fa and Meishan Zhao     

(47)

For sample rocks described by a hexagonal crystal system, the anisotropic parameters are given in Table 3. We have the calculations and presented the results in Figures 11-19. The solid lines are  and  versus  from equations (46)-(47), and the dashed lines are  and  versus  . Figures 11-19 show that the polarization direction of the plane wave propagating inside hexagonal system solids may deviate from the phase velocity direction and/or energy velocity direction. It may polarize in the clockwise or the counter-clockwise direction. The magnitudes of the deviating angles are determined by the physical and anisotropic parameters of the hexagonal system solids, as well as the propagation direction of a P-wave.

Figure 11.  and  for T-Sandstone.

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Figure 12.  and  for M-Mudshale.

Figure 13.  and  for MI-Sandstone.

33

34

Lin Fa and Meishan Zhao

Figure 14.  and  for MC-Sandstone.

Figure 15.  and  for Q-Crystal.

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Figure 16.  and  for M-shale.

Figure 17.  and  forM-sandstone-1.

35

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Lin Fa and Meishan Zhao

Figure 18.  and  for M-sandstone-2.

Figure 19.  and  for M-sandstone-3.

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4.2. Elliptical-Polarization Trajectory One way to understand the polarization states is to examine the system with respect to its elliptical-polarization trajectory. As we discussed in the last section, a VIT medium can be modeled by a hexagonal solid-state system with an elastic stiffness tensor as given in Eq. (13). In this section, we examine the VTI media interface and derive an analytical expression of the elliptical-polarization trajectory. Let’s consider a few rock samples whose mechanical property can be described by stiffness matrices of hexagonal crystal system solids, as given in Table 4, e.g., anisotropic shale (A-shale), Taylor sandstone (Tsandstone), and oil shale (O-shale). Specifically, for interfaces between these VTI media, we are going to examine two interface models: Model-1 is an interface between A-shale and T-sandstone; Model-2 is an interface between A-shale and O-shale. Table 4. Anisotropic-parameters of VTI-media: anisotropic shale (A-shale), oil-shale (O-shale), and Taylor sandstone (T-sandstone) Medium

 ( n ) (m/s)

 ( n ) ( g / cm3 )

 (n)

 *(n)

 (n)

1508 1829 2539

2.340 2.500 2.370

0.103 0.110 0.200

-0.070 -0.127 0.000

0.345 0.255 0.145

 ( n ) (m/s)

A-shale 2745 T-sandstone 3368 O-shale 4231

4.2.1. Right Rotational Elliptical-Polarization 2 2 Because VTI media are transversely isotropic, we have S x   S y  . The

system is then simplified to a two-dimensional system and the inhomogeneous refracted waves can be written by









2 2 2 2 0 S x   R  ux  exp a1  z exp i t  k   x sin  (0)   (2)   

(48)

    2 2 2 2 0 S z    R   u z  exp a1  z exp i  t  k   x sin  (0)   (2)    2   

(49)





38

Lin Fa and Meishan Zhao The phase shift of the inhomogeneously refracted P-wave is then



(2)

 Im  R (2)     arctan   Re  R (2)  

(50)

The real parts of the waves in equations (48)-(49) provide the instantaneous expressions



2



2

 

Sx   R  ux  exp a1  z cos t  k   x sin  (0)   (2) 2

2

2

0

 

Sz   R  uz  exp a1  z sin t  k   x sin  (0)   (2) 2

2

2

0





(51)

(52)

Equivalently, these two equations can be rewritten as  2 S x    | R  2 | u  2 exp  a  2 z xm 1 



 2 S z    | R  2 | u  2 exp a  2 z zm 1 



2



   cos 2 (t  k  0 x sin  (0)   (2) )  



   sin 2 (t  k  0 x sin  (0)   (2) ) .  

(53)

2

(54)

A simple algebraic recombination of these equations yields a geometric ellipse 2  S x 2   S z 2   2  2   2     2    R exp a1 z   ux   uz  2

where

2





(55)

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u  (2) x

u  (2) z

2

2

39

2 (2) 13(2) 33   v1(2)      2 (2) *  (2) (2) 2  (2)  (2)  13  33   v1    13 11   v1(2)  

2 13(2) 11(2)   v1(2)      2 (2) *  (2) (2) 2  (2)  (2)  13  33   v1    13 11   v1(2)  

The angle of the elliptical-polarization rotation is then defined as the angle between the extremity of an elliptical-polarization trajectory and the x-axis  S z 2   2    Sx   (2) (2) (2) 2  13 11   v1   arctan  2 (2) (2)  13 33   v1(2)  

 (2)  arctan 

 

 

1/ 2



tan t  k   x sin  (0)   (2) 0



     (56)

Equations (55) and (56) show that this refracted inhomogeneous Pwave is a right-rotational elliptical-polarized wave, i.e., it rotates in counter-clockwise direction with a right rotational elliptical trajectory. By rotation of the elliptical-polarization trajectory around y-axis and with increasing distance (z) from the interface, a three-dimensional polarization elliptical-sphere (e.g., Poincaré sphere-like) can be obtained. For an interface between A-shale and T-sandstone (Model-1), the system does not have an anomalous incident-angle and a critical-incidentangle is located at c(2)  48.34 . In this case, its polarization elliptical-sphere as a function of incident-angles is presented in Figure 20. The top penal shows that the size of the elliptical-polarization decreases with respect to the incident-angle. The lower penal shows the absolute value of the elliptical-polarization rotational angle at its initial state.

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Lin Fa and Meishan Zhao

Figure 20. Polarization state of an inhomogeneously refracted P-wave vs. incidentangle on A-shale/T-sandstone interface (Model-1; Case 1): (a). elliptical-polarization trajectory with a 2- dimensional map in the top showing elliptical-polarization rotation direction; (b). initial state of the elliptical-polarization rotation angle at the coordinate origin.

4.2.2. Left Rotational Elliptical-Polarization The interface between A-shale and O-shale (Model-2) has a critical incident-angle c(2)  32.99 and an anomalous incident-angle a(2)  62.16 . For incidences prior to the anomalous-incident-angle, the discussion of the elliptical-polarization state of the inhomogeneous refracted P-wave is the same as section 4.2.1.

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41

In a region after anomalous-incident angle, the transversely isotropic nature of the rock media yield two independent components of the inhomogeneously refracted P-wave, which can be expressed as     2 2 2 2 0 S x   R   u x  exp a2  z exp i  t  k   x sin  (0)   (2)    2   











(57)



(58)



(59)



(60)

2 2 2 2 0 S z    R  uz  exp a2  z exp i t  k   x sin    (2)   

The instantaneous expressions are then



2



2

 

0

 

0

Wx    R  ux  exp a2  z sin t  k   x sin  (0)   (2) 2

2

2

Wz    R  uz  exp a2  z cos t  k   x sin  (2)   (2) 2

2

2

An algebraic combination of equations 59 and 60 yields an ellipse 2  Wx 2   Wz 2   2  2   2     2    R exp a2 z   ux   uz  2

2





where

u  (2) x

u  (2) z

2

2

2 (2) 13(2) 33   v2(2)      2 (2) *  (2) (2) 2  (2)  (2)  13  33   v2    13 11   v2(2)  

2 13(2) 11(2)   v2(2)      2 (2) *  (2) (2) 2  (2)  (2)  13  33   v2    13 11   v2(2)  

The elliptical-polarization rotational-angle is then given as

(61)

42

Lin Fa and Meishan Zhao

 (2)

1/ 2 2  (2)  (2)  13 11   v2(2)          0 (0) (2)   (62)  =arctan  tan   t  k x sin      2 (2)  (2) (2)  2   13 33   v2      





Figure 21. Polarization state of an inhomogenously refracted P-wave vs. incident-angle on A-shale/O-shale interface (Model-2): (a). elliptical-polarization trajectories with 2-dimensional maps in the top showing elliptical-polarization rotation directions: top left for [  (0)  (c(2) , a(2) ) ] and top right for  (0)  ( a(2) ,90 ) ; (b). initial state of the elliptical-polarization rotation angle at the coordinate origin.

The elliptical-polarization state presented here, in the form of elliptical polarization trajectory, as a function of incident-angle is presented in

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43

Figure 21, which shows that the polarization trajectory decreases with respect to the incident-angle. It also shows that the elliptical-trajectory rotates in clockwise direction with time. Therefore, this refracted inhomogenously P-wave is a left-rotational elliptical-polarized wave.

Figure 22.  and  with respect to  for M-Mudshale. *

4.3. Effect of Anisotropy on Polarization To illustrate the effect of anisotropy on polarization, we select an example of the hexagonal system solids in Table 3, e.g., M-Mudshale. Figure 22 gives the deviations of polarization angle with respect to

44

Lin Fa and Meishan Zhao

anisotropic parameter  * = -0.25 to 0.25. Figure 23 presents the results with respect to anisotropic parameter  = -0.1 to 0.1. Figures 22-23 indicate that the polarization of the plane wave is not only dependent on its propagation direction (  or  ) but also the anisotropic parameters  and  . The *

polarization angle  as a function of  and  * is presented in Figure 24. The symbol  indicates the polarization direction of P-wave for a degenerating pure P-wave. Apparently, the anisotropy of this M-Mudshale solid influences significantly the polarization states of a P-wave.

Figure 23.  and  with respect to  for M-Mudshale.

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Figure 24. Polarization angle



as a function of



and

45

*.

Figure 25. Modulus and phase angle of refraction coefficient versus incident angle for interface between A-shale and T-sandstone.

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Lin Fa and Meishan Zhao

Figure 26. Modulus and phase angle of refraction coefficient versus incident angle for interface between A-shale and O-shale.

4.3.1. The Amplitude and Initial Phase Angle of Refraction Coefficient Now, let’s examine once again the amplitude of refraction coefficient for the VTI interfaces with media parameters presented in Table 4. Figures 25 and 26 show the P-wave to P-wave refraction coefficient as a function of incident-angle. The calculated results show that in the region after the critical incident angle, the modulus of R (2) decreases with increasing  (0) and the absolute value of its phase shift angle  (2) increases with respect to the incident-angle. 4.3.2. Effect of Anisotropy on Elliptical-Polarization State To analyze the anisotropic effect on the elliptical-polarization states of an inhomogeneously refracted P-wave, we may look at the influence from each individual anisotropy parameter, i.e.,  (1) ,  *(1) ,  (2) and  *(2) as a

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47

variable. Each anisotropy parameter is varied near the physically measured values, as given in Table 4. The calculated polarization trajectories are shown in Figures 27-29. The two-dimension maps at the top show the rotational direction of the elliptical-polarization trajectory. These figures show that the anisotropy of both incidence and refraction media influences the polarization state significantly, not only the size of the trajectory and the initial phase of the elliptical-polarization but also the shape of the trajectory. For the interface between A-shale and T-sandstone and at incidentangle  (0)  55.05 , the effects of  (1) and  (2) on the size, shape and initial phase angle of the elliptical-polarization are greater than the effects of

 *(1) and  *(2) on those of the inhomogeneously refracted P-wave is polarization wave (see Figure 27).

elliptical-polarization. The a right-rotation elliptical-

Figure 27. Influences of anisotropic parameters on the elliptical-polarization states for interface between A-shale and T-sandston, where  (0)  55.05  c(2) . (a-1 & a-2): varying  (1)  (0.04,0.2) ; (b-1 & b-2): varying  *(1)  (0.1,0.1) ; (c-1 & c-2): varying  (2)  (0.05,0.2) ; (d-1 & d-2): varying  *(2)  (0.167,0.167) .

48

Lin Fa and Meishan Zhao

Figure 28. Effect of anisotropy parameters on the elliptical-polarization status for interface between A-shale and O-shale, where  (0)  34.58  c(2) but smaller than  a(2) . (a-1 & a-2): varying  (1)  (0.1,0.2) ; (b-1 & b-2): varying  *(1)  (0.08,0.08) ; (c-1 & c-2): varying  (2)  (0.1682,0.2136) ; (d-1 & d-2): varying  *(2)  (0.02,0.02) .

Figure 29. Effect of anisotropy parameters on the elliptical-polarization trajectory for interface between A-shale and O-shale, where  (2)  63.91   a(2) (Model 2). (a-1 & a-2): varying  (1)  (0,0.12) ; (b-1 & b-2): varying  *(1)  (0.08,0.08) ; (c-1 & c-2): varying  (2)  (0.1937,0.5) ; (d-1 & d-2): varying  *(2)  (0.01,0.01) .

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49

For the interface between A-shale and O-shale, the anisotropic effects are shown in Figures 28-29. At the incident-angle of   34.58 the inhomogeneously refracted P-wave is a right-rotation elliptical(0)

polarization wave. At incident-angle of  (0)  63.91 it is a left-rotation elliptical-polarization wave.

5. ACOUSTIC-LOGGING AND SEISMIC EXPLORATION 5.1. Acoustic-Logging For acoustic petroleum logging and seismic exploration, an acoustic measurement network is essential for successful applications, e.g., prediction of new oil and gas reservoirs. Whereas, a good understanding of anisotropic effects would help in developing new technologies for such a network which can be used to obtain accurate acoustic-logging information, such as propagation velocity, signal amplitude, wavelet phase, and frequency spectrum of acoustic-logging signal in the formation.

5.1.1. Relationship between Driving-Voltage and Radiated Acoustic-Signal In this subsection we are concerned with the relationship between driving-voltage signal and acoustic-signal radiated from a sourcetransducer, and between the acoustic-signal and the electric-signal converted by a receiver-transducer. Let’s consider a piezoelectric spherical thin-shell transducer with acoustic-electric and electric-acoustic conversions. Two equivalent circuits are shown in Figure 30. In solving piezoelectric and particle equations of motion, the electric-acoustic and acoustic-electric impulse response functions can be obtained, which can be written as h1  t   K1e1t  K 2 e 1t cos 1t  1 

(63)

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Lin Fa and Meishan Zhao

h3  t   K3e3t  K 4 e 3t cos 3t  3 

(64)

where K 1 , K 2 , K 3 and K 4 are constants; 1 and  3 are the damping coefficients for direct current; 1 and  3 are damping coefficients for alternate current; 1 and 3 are the phase shifts; and 1 and 3 are the center frequencies respectively.

of

source-transducer

and

receiver-transducer,

Figure 30. Two equivalent circuits of the transducers: (a) the source-transducer; (b) the receiver-transducer.

Table 5. Physical and geometrical parameters of transducer material PZT-7A and acoustic impedance of the coupling medium 

d

3

(kg / m )

(10

7600

-60

31 12

m/V)

T  33

(10 -9 F/m)

s12E (10 12 m 2 /N)

rb Zm s11E 2 6 2 (10 m /N) (10 kg/m  s) (cm)

lt (cm)

3.7613

-3.2

10.7

0.8

12

1.2205

8

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51

Let’s consider a thin-shell transducer which is composed of piezoelectric material PZT-7A. Table 5 lists its physical and geometrical parameters, and  is the material density of the transducer; d31 ,  33T , s12E and s11E are the piezoelectric, dielectric and strain constants; Z m is the acoustic-impedance of coupling medium around the transducer; rb and lt are the average radius and the shell thickness of the transducer. The coupling medium around the transducer is transformer-oil and the output impedance of the driving circuit is taken to be 50  .

Figure 31. Driving-voltage signal: (a) the waveform; (b) amplitude spectrum.

Figure 32. Radiated acoustic signal: (a) the waveform; (b) amplitude spectrum.

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Lin Fa and Meishan Zhao

Figure 33. Electric-acoustic conversion of source-transducer: (a) electric-acoustic impulse response; (b) amplitude spectrum.

Figure 34. Acoustic-electric conversion of receiver-transducer: (a) acoustic-electric impulse response; (b) amplitude spectrum.

As shown in Figure 31, we use a gated sine voltage signal to excite the source-transducer. Acoustic signal radiated by the source–transducer is

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53

shown in Figure 32. It is noted, the driving-voltage convolution with the electric-acoustic impulse response from the transducer may also be used as a signal source. This may make the acoustic logging forward model much closer to actual acoustic logging. From equations. (63)-(64), the electricacoustic and acoustic-electric conversions in time- and frequency-domains are presented in Figures 33-34.

5.1.2. A Model Acoustic-Logging Transmission Network Now, let’s consider the geometrical configuration of an acoustic logging system, shown in Figure 35. A logging tool is placed in a fluidfilled cylindrical borehole, which has a radius “a”. Electric driving signal excites source-transducer to emit an acoustic signal. This acoustic signal propagates to a receiver-transducer via media in borehole and around borehole. Then, the acoustic signal is converted into an electric signal by receiver-transducer and recorded by the logging tool. The impact of a transducer on the amplitude and frequency of a logging signal can be significant. This is mostly the problem of electricacoustic and acoustic-electric conversions. Nevertheless, most of the traditional acoustic-logging methods simply neglect the transmission time delay caused by these conversions which leads to logging errors. To establish an improved model, we must realize that, theoretically, the travel time of an acoustic-logging signal in media must be the propagation time measured by a traditional logging device minus transmission time delay. Based on the theory of signal transmission, an acoustic-logging transmission network (ALTN) model can be established, as given in Figure 36. Eliminating the flaws embedded in traditional acoustic logging models, the new model network takes a full consideration of logging signal conversion, including transmission time-delay caused by the transducers. In turn, the new ALTN model yields a more realistic measurement on propagation speed, signal amplitude, and frequency of the logging signals. The geometrical configuration of such a network model is described and is shown in Figure 35.

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Lin Fa and Meishan Zhao

Table 6. Physical parameters of the borehole fluid and MC-sandstone around borehole Medium

  kg/m 3 

 p  m/s 

 sv  m/s 

Borehole fluid Formation

1.2 2.16

1540 5943

3200

Figure 35. A geometrical configuration of an acoustic-logging system: T and R are the source-transducer and receiver-transducer separated by a distance L.

Figure 36. A model acoustic-logging transmission network (ALTN): the drivingvoltage signal u1  t  and the measured logging signal wavelet u3 (t ) are defined as the input and output respectively; x(t)as the acoustic signal radiated from a receivertransducer; p (t , z0 ) is the acoustic signal wavelet propagating to a receiver-transducer.

In a logging process, we may consider receiver-transducer as an electric-acoustic filter, the propagation media as an acoustic filter, and receiver-transducer as an acoustic-electric filter. The measured signal is

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55

summarily influenced by various factors, e.g., physical properties of the transducers, the physical and geometrical properties of the propagation media, etc. To emphasize the transmission time delay caused by acoustic-electric conversion of receiver-transducer, we use a Tsang wavelet [33] as an example for the acoustic-signal radiated by receiver-transducer x(t )  4 tet sin(0t ) H (t )

(65)

The parameter  is a damping coefficient and 0 is the center frequency of the wavelet. It may be useful, from a practical point of view, to examine the transmission time delay caused by the acoustic-electric conversion of a receiver-transducer. Using the parameters given in Table 6, the calculated results are shown in Figure 37. The waveform and amplitude of the output signal are compared to that of the normalized Tsang wavelet spectrum. Several variables and parameters are involved in calculations, where  is the density of propagation medium, vp is the acoustic speed of P-wave, and v sv is the acoustic speed of SV-wave. The Tsang wavelet parameters are 0  2 104 rad/s and   0.30 /  . The distance from sourcetransducer to receiver-transducer L = 1.3224 m, and the borehole radius a = 12 cm. Figure 37 shows that for the acoustic-logging signal used in this example, the transmission time-delay caused by acoustic-electric conversion of receiver-transducer is not small, up to 3.1886 μs. Comparing to the negative wave amplitude of p(t,z) in Figure 37(c), the wave amplitude of u3 (t ) in Figure 37(e) has initially a relative decline, then changes direction to a positive region. This shows that the effect of acoustic-electric conversion from receiver-transducer on logging signal cannot be simply ignored.

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Lin Fa and Meishan Zhao

Figure 37. Acoustic-electric conversion of receiver-transducer on the measured logging signal: (a)-(b). the normalized Tsang wavelet spectrum for waveform and amplitude; (c)-(d). those of acoustic signal p(t, z0) reaching at receiver-transducer via the borehole fluid and the formation around the borehole; (e)-(f). the electric signal u3 (t ) converted by receiver-transducer.

If the head-wave amplitude of the acoustic-logging signal is small, the cement bond quality of the cased-well is defined positive; otherwise, the

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57

cement bond quality must be considered as poor. Practically, if the impact of acoustic-electric conversion on the head-wave amplitude is not properly treated, a bad cementation quality may be misjudged as a good one and vice versa. The discussed ALTN model may be extended to a system with multiple transducers, i.e., an array acoustic-source (AAS) in the borehole and an array acoustic-receiver (AAR). These two ALTN models are called addition ALTN models. The excitation time delay t between two neighboring transmitting elements can be calculated from

t  d / v

(66)

where d is the distance between two neighboring transmitting elements and v is the acoustic velocity of P-wave in the formation around the borehole. It has been known that an array acoustic-source (AAS) with a single receiver is reciprocal to that of a single source with an array acousticreceiver (AAR). Consider an extended ALTN network to include N receivertransducers and M receiver-transducers. P-wave velocity around the borehole is used to adjust the AAS excitation time-delay. The acoustic signals emitted by all transmitting elements have the same phase in AAS propagating around the borehole, then to reach the receiving elements in AAR. The time-delay between acoustic-logging signals in reaching receiving elements may be adjusted by equation (66). The amplitude of the stacked head-wave is roughly proportional to both the number of sources and the number of receivers, i.e., proportional to the product of M and N. This extended ALTN model is known as a multiplication ALTN. Considering, as an example, of N = M = 4, i.e., AAS and AAR consist of four transducers each, we set the interval between two neighbor transducers to be 82 mm and the distance from AAS to AAR to be 2.44 m. The calculated acoustic beam directivities for this ALTN are presented in Figure 38. Clearly, the acoustic-beam steering efficiency of the multiplication ALTN is much higher than that of the addition ALTN.

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Lin Fa and Meishan Zhao

Nevertheless, both networks increase greatly the head wave amplitude for the acoustic-logging signal.

Figure 38. Acoustic-beam steering directivities of addition and multiplication ALTNs: (1) the directivities of the two addition ALTNs; (2) the directivities of the multiplication ALTN.

5.1.3. Acoustic Signal Propagation in Drill-Collar Acoustic logging while drilling is an important application, which is mainly used for acoustic velocity measurement around oil wells, e.g., horizontal wells, deviated wells on land, and cluster wells on offshore. Conventional logging method can be applied to acoustic-logging tool while drilling. Acoustic probes may be used as the general acoustic-transducers or multi-pole acoustic-transducers. The acoustic-logging signals are measured, acquired and stored by the electric circuit module in the acoustic logging tool while drilling (ALTWD). Then, the logging data could be processed by a proper software program and viewed with a computer interface. The drill-collar is usually made of steel. The outer-shell of ALTWD is a steel- casing with multiple grooves. The ALTWD is placed in drill collar and the outer-shell of ALTWD is scheduled for torsion creation during

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59

drilling. Bearing huge torsion-force, the drill collar-cannot be grooved, or else it could be ruined.

Figure 39. A schematic representation of the propagation paths of an acoustic signal.

As shown in Figure 39, there are small windows on the drill-collar and on the outer-shell of ALTWD which are in the vicinity of sourcetransducers and receiver-transducers. The acoustic signal radiated by a source-transducer in ALTWD may pass through the pipe layers via the windows, to reach the formation around the borehole. The acoustic-signal coming from the formation may be collected by receiver-transducers in ALTWD. One of the technical difficulties with ALTWD is that the propagation speed of P-wave in steel is typically about 5900 m/s. This speed is usually larger than that of low-velocity/intermediate-velocity formation. The propagation path of the acoustic signal in drill-collar is shorter than that of formation. Without a specific design, the acoustic signal from the drillcollar may reach receiver-transducer unwittingly ahead of the signal from the formation around the borehole. To make a specific design, we note that scattering takes place when acoustic-signal impinges on the interface between two different media.

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Lin Fa and Meishan Zhao

Applying acoustic-scattering theory and medium acoustic-absorption technology, the outer-shell of drill-collar can be designed properly with required specifications. By doing so, the outer-shell ensures that acoustic signal emitted by the source-transducer would pass fully through the windows on outer-shell of ALTWD, as well as on drill collars. Additionally, some technical measures and special algorithms are used to process these logging data. This would ensure the obtained acoustic logging signals come from the formation around the borehole, rather than from drill-collar.

5.2. Reflection/Transmission between Two Anisotropy Rock Slabs 5.2.1. Application of Reflection/Transmission to Seismic Exploration Amplitude variations with offset (AVO) is one of the most important analyses in studies of reflection coefficients. P-wave amplitude variation with respect to an incidence angle is affected by acoustic impedances of Pand SV-waves on both sides of the reflector. AVO anomalies can be used to predict zones of Poisson’s ratio change which is a direct hydrocarbon indicator [50]. Anisotropy could also affect significantly on AVO application. Rock anisotropy is generally described as transversely isotropic, So, the presence of rock anisotropy can severely distort AVO analysis. In fact, it has been known that elastic anisotropy is so common in Earth’s interior that it is virtually impossible to avoid it in geophysical applications. At the interface between two transversely isotropic media with a vertical axis of symmetry (VTI), a set of fourth-order polynomials have been established for reflection/refraction angles B1(1) sin4  (1,3)  B3(1) sin 2  (1,3)  B5(1)  0

(67)

B1(2) sin4  (2,4)  B3(2) sin2  (2,4)  B5(2)  0

(68)

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Once a P-wave impinging on the interface between two VTI media, in terms of the reflection/refraction coefficients, the relationship between displacement and traction across boundary can be written as  a11   a21  a31   a41

a12 a22 a32 a42

a13 a23 a33 a43

a14  R1   b1      a24  R2   b2   a34  R3   b3      a44   R4   b4 

(69)

where we adopt the superscripts {m} = {0, 1, 2, 3, 4} for incident Pwave/SV-wave (m = 0), reflected P-wave (m = 1), refracted P-wave (m = 2), reflected SV-wave (m = 3), and refracted SV-wave (m = 4). R(1) is the P-wave to P-wave reflection coefficient, R(2) is the transmission coefficient of P-wave to P-wave refraction, R(3) is the P-wave to SV-wave reflection coefficient, and R(4) is the transmission coefficient of P-wave to SV-wave refraction. Explicitly, the matrix elements are a11  ux(1) , a12  ux(2) , a13  uz(3) , a14  uz(4) ,

a21  uz(1) , a22  u z(2) , a23  ux(3) , a24  uz(4)

a31 

(1) (1) c13(in ) u x(1) sin  (1)  c33 u z cos  (1) v(1) ( (1) )

a32  

( re ) (2) c13( re) u x(2) sin  (2)  c33 u z cos  (2) v (2) ( (2) )

a33  

( in ) (3) c13(in ) u z(3) sin  (3)  c33 u x cos  (3) v (3) ( (3) )

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Lin Fa and Meishan Zhao a34  -

c33(re ) ux(4) sin (4) + c13(re ) uz(4) cos (4) v(4) ( (4) )

a41 

( in ) c44 (u x(1) cos  (1)  u z(1) sin  (1) ) v(1) ( (1) )

a42 

( re ) c44 (u x(2) cos  (2)  u z(2) sin  (2) ) v (2) ( (2) )

a43  

( in ) c44 (u x(3) sin  (3)  u z(3) cos  (3) ) v (3) ( (3) )

a44  

c44(re ) (uz(4) cos (4)  ux(4) sin (4) ) v(4) ( (4) )

The elements of vector b are b1  ux(1) , b2  uz(1)

b3  

b4 

( in ) (1) c13(in ) u x(1) sin  (1)  c33 u z cos  (1) v(1) ( (1) ) 已改

( in ) c44 (u x(1) cos  (1)  u z(1) sin  (1) ) v(1) ( (1) )

By the anisotropic parameters for T-sandstone and O-shale, as listed in Table 4, the calculated reflection coefficients are presented in Figure 40. These results can be used for AVO analysis in seismic exploration.

Recent Progress in Acoustical Theory and Applications

Figure 40. Reflection/transmission coefficients versus θ and ε(re): ϕ(m) is the phase of R(m), m = 1, 2, 3, 4.

63

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Lin Fa and Meishan Zhao

5.2.2. Inverse Oil Reservoir by Logging and Seismic Data With respect to the various possible geological structures, e.g., thinout, top-lap, down-lap, etc., a seismic wavelet dictionary can be established by invoking logging data from multiple oil wells in a given region. A match pursuit can be performed for words in the dictionary with the seismic exploration data. The reflectivity series of underground formation can be obtained from measured seismic wavelets and the words from the seismic wavelet dictionary. By observing amplitudes and phases of the obtained reflectivity series, an oil reservoir may be revealed. This technology can be processed in two steps: (i) creation of a seismic wavelet dictionary, and (ii) software processing of reflectivity series inversion.

Figure 41. A geological structure of a thin-out.

Recent Progress in Acoustical Theory and Applications

Figure 42. A geological structure of an inter-bed.

Figure 43. A geological structure with two down-laps.

65

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Lin Fa and Meishan Zhao

In calculations of the formation reflectivity series, a software program is used to scan all words in the seismic-logging dictionary with every actual measured seismic signal wavelet. This scan would locate the maximum correlation coefficient. According to the shift-invariance norm, iterative calculations are performed by utilizing a match pursuit algorithm to achieve a proper convergence. When the calculation is converged, a minimal residual between the “words” with a maximal correlation coefficient is obtained. Then, the trace of the actual seismic signal wavelet is reached. For example, from the seismic wavelet dictionary, words in Figure 41 show a geological structure of a thin-out. Figure 42 presents a geological structure of an inter-bed, and Figure 43 reflects a geological structure with two down-laps.

5.3. Cylindrical-Shell Transducer and Slim-Hole Acoustic-Logging Tool In this section we discuss the directivity of cylindrical-shell transducers on acoustic-beam steering efficiency for slim-hole logging tool.

5.3.1. Electric-Acoustic Conversion Network Consider a cylindrical-shell transducer with shell thickness lt, radius rb and height H, which is composed of piezoelectric material PZT-5H whose density and physical and piezoelectric parameters are shown in Table 7. Transformer oil can be used as a coupling medium around the transducer in the logging tool. There, the acoustic velocity and density are given by vm = 1425 m/s and ρm = 856.5 kg/m3. The logging-tool transducers, both the source and receiver transducers, and the transformer oil are isolated from the borehole fluid with a thin rubber pocket. The created acoustic reflection and attenuation, caused by robber pocket, are small. Then, a rubber pocket can be considered as an acoustic penetrating

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67

layer. The acoustic velocity in the borehole fluid is approximately the same as that in transformer-oil. The directivity of the transducer in the transformer oil can then be approximated by the directivity in the borehole fluid, which can be expressed as  2  2 rb  2  2 rb  J 0   cos    J1     G ( )     2  2 rb  2  2 rb J0    J1        

1/2

  cos   cos 2      2  cos     

H  sin  sin      H sin 

(70)



where J 0 and J1 are the zeroth-order and the first-order Bessel functions,

 is the wavelength of acoustic wave in coupling fluid, and  is the angle between the propagation direction of the acoustic signal radiated by the cylindrical transducer and the direction vertical to the transducer’s sidewall.

Figure 44. The upper penal is an electric-acoustic transmission network (source). (b). Acoustic-electric transmission network (receiver).

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Lin Fa and Meishan Zhao

Now, let’s consider the electric-acoustic and acoustic-electric transmission network for the cylindrical-shell transducers, shown in Figure 44. The electric-acoustic and acoustic-electric impulse responses of a cylindrical-shell transducer are written as h1 (t )  K11exp  1t   2K12 exp  1t  cos 1t  1 

(71)

h3 (t )  K31exp  3t   2K32 exp  3t  cos 3t  3 

(72)

For a source-transducer, its electric terminals are the input terminals and its mechanical terminals are the output terminals. For a receivertransducer, that process is reversed and generically 1  3 . Even though the physical and geometrical parameters are the same for both sourcetransducer and receiver-transducer, the parameters of driving-circuit are different from those of the measurement circuit. So, the electric-acoustic transmission properties of source-transducer and acoustic-electric receivertransducer are still somewhat different.

Figure 45. Configuration of acoustic-logging for slim-hole acoustic-logging tool with an array source and a receiver-transducer. z is the interval between two neighbor transducer elements in the array source and L is the distance from the transducer element T1 in array source to receiver-transducer.

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69

Consider a logging tool consisting of a receiver-transducer and an array source which has n cylindrical-shell transducers (T1, T2, …, Tn). A fluid-filled cylindrical borehole is submerged in a reservoir of elastic medium. The lowest ranking transducer T1 is set at the origin, as shown in Figure 45. The transmission function and impulse response for the acoustic signals radiated by each transducer in the array source can be written as H 2l ( )  H 2 dl    H 2 rl  

  z  (l  1)z   1 exp  i (l  1)t   z  (l  1)z vf    

=



1 (2 ) 2



exp[ i (l  1)t ]  A(k ,  ) exp ik z [ z  (l  1)z ]dk z

(73)



and 

h2l (t )   H 2l ( ) exp[it ]d

(74)



where k z is an axial wavenumber, t is the time delay between two neighboring transducers in the array source, [ z  (l  1)z ] is the distance from the lth transducer-element in array source to receiver-transducer, and Vf is the acoustic velocity in borehole fluid. Logging information are acquired from acoustic-logging process, such as the head-wave amplitude, acoustic velocities of P-wave and S-wave. Where, S-wave is shear wave which stands for either SV-wave or SH-wave. The direct wave is absorbed by the sound absorption body in the logging tool. As such, in calculations of the acoustic-beam steering, the direct wave is mostly omitted. For the configuration of acoustic-logging, as given in Figure 45, a flowchart of the acoustic-logging transmission network in the time-domain is shown in Figure 46. The output signal, i.e., the electric output from the receiver-transducer can be expressed as

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Lin Fa and Meishan Zhao n



U 3  t    U1l t   l  1 t  H t   l  1 t  l 1



* h1l  t ,(l  1) z  G1l (c )  * h2  t , z  * G3 (c )h3 t , z 

(75)

Figure 46. A schematic diagram of an acoustic-logging transmission-network.

In equation (75), H(•) is Heaviside unit step function, c is the critical incident angle from the borehole fluid to the formation around the borehole, G1l (c ) is the value of the directivity coefficient of each transducer-element in the array source, and G3 (c ) is the directivity coefficient of the receiver in the opposite direction of c . The transducer elements in the source and the receiver-transducer have the same geometrical and physical parameters. There is also a reciprocity of the radiating and the receiving directivities of the cylindrical transducer. So, we have G11 (c )  G12 (c )  ...  G1n (c )  G3 (c )  G(c )

and

(76)

Recent Progress in Acoustical Theory and Applications n



71



U 3  t    U1l t   l  1 t  H t   l  1 t  *  h1l  t ,(l  1) z   l 1

* h2  t , z  * h3  t , z  G ( c )

(77)

2

where G2 (c ) describes the effect of radiating and receiving directivity of the slim-hole logging tool with an array source and a receiver on the acoustic-logging signals, which is called the directivity weighted coefficient.

5.3.2. Effects of Directivity on Acoustic-Logging Signals Consider a transducer which is composed of piezoelectric material PZT-5H. Its shell thickness is one-tenth of its outer-diameter. On the geometrical-size of slim-hole, its maximal diameter of the cylindrical transducer is usually not more than 30 mm. The compliant, dielectric, and piezoelectric constants of PZT-5H, and the acoustic impedance of the around transducers through the medium are given in Table 7. The loading center frequencies of electric-acoustic and acousticelectric conversions are influenced by the piezoelectric parameters and the geometrical size of the cylindrical transducer, such as the average radius ( rb ) of the transducer. The general tendency of loading center frequency versus average radius is shown in Figure 47. Table 7. Physical parameters of piezoelectric material PZT-5H and acoustic impedance of coupling fluid around the transducers s11E (m2 /N)

11T (F/m2 )

d31 (m/V)

 (kg/m3 )

Zm (kg/m2  s)

16.5  1012

3.009  108

2.74  1010

7500

1.221106

At rb = 15 mm, the f1 (rb ) has a value of 29.019 kHz. For this geometry of the slim-hole acoustic-logging tool, in varying height of the cylinder, the directivities of the cylindrical-shell transducer from equation (70) is shown in Figure 48. It shows that the longer the cylinder, the more concentrated

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of the radiated acoustic energy in the direction vertical to the wall of the transducer. For example, at a cylinder height of 25 mm, the acoustic radiation area is 2359.5 mm2. Conventional acoustic-logging tools usually have a height of 48 mm and an outer diameter of the transducer and 52 mm. The acoustic radiation area is 7841.4 mm2, which is more than 3 times of the former.

Figure 47. Relation of load center frequency f1 versus average radius

rb

Figure 48. Directivity of cylindrical-shell transducer with different height (H) of the cylinder at rb = 15mm.

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To strengthen the received acoustic signal and increase the radiated acoustic energy, an adopting phase control technology with respect to line array-source may be applied. This technology increases the effective radiation area of the acoustic source, improves the radiating and receiving directivity of the slim-hole logging tool, and makes the radiated acoustic energy more concentrated in the direction of the critical incidence angle. If we set all components of acoustic signals propagating in the formation around borehole to stack with the same phase, the amplitude of the received acoustic signal will be at its maximum. The excitation time delay between neighboring transducers in the line array source is then t0  z / v2

(78)

where v2 is P-wave (or S-wave) velocity of the formation around the borehole. Now, let’s examine the maximal head-wave amplitude varying with respect to physical parameters of the cylindrical-shell transducer. Let’s consider a gated sinusoid wave as driving-voltage signal for calculation U1 (t )   H (t )  H (t  t1 ) sin 0t

(79)

where, t1 is the window width of the sinusoid wave and 0  2 f0 is the angular

frequency.

We

selected the signal parameters f0  f1  29.019 kHz , t1  6 / 1 , R  0.135 m , L  1.4 m , and

z  40 mm , and the parameters of borehole fluid and formations around borehole as presented in Table 8. Start with the driving-voltage signal excites only the first transducer element T1 in the array source, as shown in Figure 45. The calculated headwave amplitude of received acoustic-logging signal varying with respect to the height of the transducer are shown in Figure 49. The head-wave amplitude per unit radiation area is defined as the ratio of received headwave amplitude to acoustic radiation area of the transducer.

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Lin Fa and Meishan Zhao Table 8. Borehole fluid density, P-wave and SV-wave velocities in formation around borehole Medium

 (kg/m3 )

v p (m/s)

vs (m/s)

Borehole fluid Formation 1 Formation 2 Formation 3

880 2100 2160 2400

1440 2770 3500 5640

1630 1900 3200

Within a unit radiation area of a transducer element, the received headwave amplitude decreases monotonically with increasing the length of the transducer. For the entire radiation area of the transducer element, the received maximal head-wave amplitudes occur at an optimal transducer height. For the case of formation 1, at a transducer height H = 0.0350 m, the normalized head-wave amplitude reaches its maximum which is equal to 0.5105; For formation 2, at H = 0.0440 m, the normalized head-wave amplitude has its maximum 0.6281; and for formation 3, at H = 0.0706 m, the normalized head-wave amplitude is equal to 1, which is its maximal value.

Figure 49. The maximal head-wave amplitude (normalized) obtained from the three kinds of formations shown in Table 2: curves (1) and (4) are the calculated results for formation 1; curves (2) and (5) are those for formation 2; and curves (3) and (6) are those for formation 3.

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Figure 50. The head-wave amplitude varying with respect to excitation signal.

Figure 49 shows that the larger the acoustic velocity of the formation, the larger the obtained head-wave amplitude. For a given radius, a larger transducer height (or radiation area) cannot guarantee a larger head-wave amplitude. It means that the acoustic energy radiated by the transducer with larger height is more concentrated in the direction vertical to the wall of the transducer. The formation with larger acoustic velocity has a smaller incident critical-angle.

5.3.3. Head-Wave Amplitudes and Excitation Delay For a slim-hole logging tool with a line array source and a receiver, let’s examine the relationship between the received head-wave amplitude and the excitation delay, with and without neglecting the transducer directivity. Consider a line array source with four transducer elements. The calculated relationships between received head-wave amplitudes and excitation time delay are given in Figure 50. For the three formations in Table 8, the lines (1)-(3) are the cases of neglecting transducer directivity and the lines (4)-(6) are the cases including transducer directivity. Figure 50 shows that the received head-wave amplitudes for the cases with directivity are smaller than those of neglecting the directivity. The

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directivity has greater effect on logging signals for slow velocity than that of fast velocity formation. This may be understood as that the slower velocity formation has a greater incidence critical-angle. The P-waves created by all transducer elements in the source array stack with the same phase and the received head-wave amplitude yields a maximum. The head-wave amplitude of the slim-hole logging tool, as presented in Figure 50, may be converted to radiating and receiving directivity. The curves (1)-(3) in Figure 51 are the cases of formations of the media around borehole. It shows that the larger the acoustic velocity of the formation, the smaller the acoustic-beam steering angle. Because faster velocity formation around borehole yields smaller critical incidence angle, the effect of transducer directivity on the received head-wave amplitude is smaller for the fast velocity formation than that of slow velocity formation. Table 9. The number of transducer elements versus received head-wave amplitude N HWA1 HWA2

1 0.2445 0.2318

2 0.4925 0.4670

3 0.7443 0.7057

4 1.0000 0.9482

Figure 51. Radiating and receiving directivity of the logging tool with an array source and a receiver.

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Figure 52. Steered acoustic-beam width versus neighboring transducer elements: curve (1) is the radiating and receiving directivity for z =25 mm; (2) z =35 mm; and (3) z =45 mm.

It is also informative to look at the width of steered acoustic-beam with respect to the distance between two neighboring transducer elements in a line array source. Take formation 2 as an example and set z at 25 mm, 35 mm, and 45 mm, the radiating and receiving directivities are shown in Figure 52. It shows that the larger this distance, the narrower the steered acoustic-beam width, and the more concentrated the radiated P-wave energy in the direction of incident angle. Finally, let’s examine the head-wave amplitude with respect to the number of transducer elements in an array source. Take formation 3 as an example, when P-wave velocity is used to adjust the excitation delay of the source array, the number of transducer elements in the array source and the received head-wave amplitude are presented in Table 9, where HWA1 is the case of neglecting transducer’s directivity, and HWA2 is that of with transducer’s directivity. Table 9 shows that if the number of transducer elements in the array source is n, the received head-wave amplitude is approximately n-times of the received head-wave amplitude of a single transducer element.

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6. MORE DISCUSSIONS OF ACOUSTIC WAVE AT SOLID-SOLID INTERFACE 6.1. Seismic Data and Signal Analysis at VTI-VTI Media Interface In this section we provide more discussions of anisotropic effects on seismic data and signal analysis for transversely isotropic rock media with vertical anisotropy (VTI), as shown in Figure 53. These effects are significant in many practical applications, e.g., earthquake forecasting, materials exploration inside the Earth’s crust, and more.

Figure 53. Transversely isotropic elastic VTI medium with a vertical axis of symmetry.

Figure 54. The relationship between phase velocity and energy velocity in xz-plane within an isotropic media.

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In seismic explorations, time-depth relation from seismic reflection data was used for time-depth conversion, to achieve a better understanding of the geological structure of Earth’s interior. In quantitative analysis of the anisotropy effects on seismic data and signal, the accuracy of a time-depth relation is critically important, because it is closely related to the velocity analysis of seismic signals, depth estimation of reflectors, and synthesis of seismograms. For isotropic media, the wave-front is spherical, the phase velocity is perpendicular to its wave-front, the energy velocity and the density vector are in the same direction, and the direction and magnitude of the phase velocity are the same as those of the energy velocity, as shown in Figure 54. A primary wave (P-wave) is the fastest traveling wave compared to other elastic waves, e.g., S-waves and surface waves. By detecting the non-destructive P-waves, it is possible to obtain warnings of earthquakes before they arrive, because P-waves travel faster than the destructive secondary waves (S-waves). Following P-waves, SV-waves are the second type of wave to be recorded on an earthquake seismogram. In seismic applications, S-waves are polarized vertically, and are called SVwaves. The physical properties of a VTI medium is usually modeled by a stiffness matrix, as given in equation (13). The normalized displacement vectors of P- and SV-waves in infinite VTI medium can be written as  u  S p   x  exp i (t  k p  r )   u z 

(80)

 u  S sv   z  exp i (t  k sv  r )   u x 

(81)

where u x and u z are the polarization coefficients. The signs of these coefficients are dependent on corresponding phase angles and anisotropic parameters.

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Figure 55. Phase velocity and energy velocity in xz-plane within elastic VTI media.

In seismic exploration, the actual seismic wave propagates in the ray direction. A seismic wavelet does not create a frequency dispersion during its propagation in elastic VTI medium, so the phase velocities are constants for all frequency components. Meanwhile, the wave-front is no longer spherical. Both the phase velocity direction and its magnitude are usually different from those of energy velocity (see Figure 55). For a homogenous P-wave or SV-wave propagating in xz-plane, the phase velocity can be written as vi  eki

 ki



  sin i    ki  cos i 

(82)

where the subscript {i}={p, sv} stands for P-wave and SV-wave, respectively and eki is the unit vector of wave-front normal. The power density flux is * *  Pix  1 * 1  VixTi1  Viz Ti 5  Pi   Vi  Ti       *  2 2 VixTi 5  Viz*Ti 3   Piz 

(83)

The x- and z-components of power density flux are written as

Ppx 

k p

ux (c11ux sin  p  c13uz cos p )  uz c44 (ux cos p  uz sin  p )   2  (84)

Recent Progress in Acoustical Theory and Applications

Ppz 

Psvx 

Psvz 

81

k p

ux c44 (ux cos p  uz sin  p )  uz (c13ux sin  p  c33uz cos p )   2  (85)

 ksv uz (c11uz sin sv  c13ux cossv )  ux c44 (uz cossv  ux sin sv ) 2 (86)

 ksv 2

u z c44 (u z cossv  ux sin sv )  ux (c13u z sin sv  c33ux cossv ) (87)

The relationships between energy angle and phase angle are obtained as  Ppx  Ppz 

 p  tan 1 



   





 l p c11l p sin  p  c13m p cos p  m p c44 l p cos p  m p sin  p  tan 1   l p c44 l p cos p  m p sin  p  m p c13l p sin  p  c33m p cos p 







  (88)  

 Psvx    Psvz 

sv  tan 1 

 m  c m sin  sv  c13lsv cos sv   msv c44  msv cos sv  lsv sin sv    tan 1  sv 11 sv   msv c44  msv cossv  lsv sin sv   lsv  c13msv sin sv  c33lsv cos sv  

(89)

In which,  p is the angle between the wave vector of P-wave and z-axis and sv is the angle the wave vector of SV-wave and z-axis. Invoking the relationship between energy and phase velocities, energy velocity can be expressed as vei  vi / cos(i  i )

(90)

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Based on equations (80)-(81), the normalized displacement vectors can be rewritten as   x sin  p  z cos  p  u  S p   x  exp i  t  vep cos( p   p )    u z 

   

  x sin  sv  z cos sv  u  Ssv   z  exp  j  t  vesv cos(sv   sv )    ux 

(91)

  

(92)

Now, let’s consider some rock samples with well-known anisotropic parameters, as listed in Table 10. Specifically, the three samples are Msandstone with small anisotropy, C-sandstone with weak to moderate anisotropy, and M-shale with moderate to strong anisotropy. Table 10. Rock anisotropic parameters for Mesaverade sandstone, Mesaverade Calcareous sandstone, and Mesaverade shale Sample

 (m/s)

 (m/s)



*



 (g/cm3 )

M-sandstone C-sandstone M-shale

4633 5460 3377

3231 3219 1490

-0.026 0.000 0.200

-0.004 -0.345 -0.282

0.035 -0.007 0.510

2.710 2.690 2.420

When a P-wave propagates in the xz-plane with a plane reflector and phase velocity v p , energy velocity vep , and the travel-time t p , we set the horizontal distance from source to receiver i.e., offset) to be 2l. The actual time-depth relation of reflector between two homogenous rock layers is then h  {vep [ p ( p )]t p / 2}2  l 2

(93)

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For a rock with anisotropy, replacing energy velocity by phase velocity yields a time-depth relation for the reflector

hp 

v 2p ( p ( p ))t 2p 4

 l2

(94)

Furthermore, when neglecting the effect of the anisotropy by using vertical phase velocity to replace the energy velocity, the time-depth relation yields

hi 

 2t 2p 4

 l2

(95)

Now, we can calculate the actual reflector depth (h) and the induced depth errors, hp  hp  h , as a function of the reflection travel-time, t p . We can also calculate the obtained reflector depth as a function of the offset midpoint (l). Then, we can calculate the percentage errors from hk h h  k  100%, h h

k  i, p .

(96)

Table 11. Induced relative depth errors as a function of wave traveling time t p with a fixed offset ( l  2000 m ) in neglecting anisotropy Sample

t p (ms )

M-sandstone C-sandstone M-shale

508.83 1203.02 609.02

h p / h

0.2% 2.0% 14.0%

hi / h 9.1% 18.1% 52.5%

The numerical results for the selected samples are given in Table 11 and Table 12.

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Lin Fa and Meishan Zhao Table 12. Induced relative depth errors as a function of offset ( l ) in neglecting anisotropy Sample

l (m)

M-sandstone C-sandstone M-shale

2000.0 945.1 2000.0

h p / h

0.1% 2.5% 3.1%

hi / h 2.2% 20.5% 4.1%

Figure 56. Propagation path of seismic reflection signal in anisotropic medium in xy-plane.

Figure 57. A measurement line and source transversally move in a given constant interval l each time (i.e., move in y-axis direction).

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Figure 58. A measurement line rotates from OA to OA ' at the same angle interval each time with respect to the vertical axis of the source.

Consider the propagation and reflection of a wave, as shown in Figure 56, observed at position R. A wave starts from source S, propagates to point O, and then is reflected to the position R. The energy incident angle

i on the reflection point O is different from the phase incident angle i . The energy angle is determined by the offset and the reflector depth, while the phase angle is dependent on the energy angle and the rock anisotropy. Because both energy and phase velocities are functions of phase angle and the rock anisotropy, the travel-time of a reflected signal varies with respect to the system factors, e.g., the reflector depth, anisotropy, and phase and energy angles for a given offset. In seismic exploration, the data for the reflected seismic signal can be obtained in different moving patterns of a measurement line. Here, we assume that the measurement line consists of one source S and N receivers R1, R2, …, RN. All receivers are in a straight line on the surface. In case that the measurement line is parallel to the x-axis, the measurement of the acquired seismic data can be obtained in the ydirection with each increment l . Figure 57 shows a step-by-step measurement from AB to A ' B ' approximations in ignoring anisotropy.

with

various

reflections

and

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Lin Fa and Meishan Zhao

Figure 59. In neglecting the anisotropy, the relation of measurement error hi versus both t p and coordinate y.

Figure 60. In neglecting the anisotropy, the relation of reflector depth hi versus both the measurement length l and coordinate y.

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Figure 61. The depth error due to neglecting the anisotropy rotating with respect to vertical axis. θ is the rotation angle of the measurement line.

Figure 62. Reflector depth error created by rotating measurement line with respect to vertical axis.

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Suppose that the measurement line rotates around the source S at a fixed-angle interval and the measurement line acquires seismic data with each rotation. Figure 58 demonstrates the rotation of the measurement line from OA to OA ' . Figures 59-60 show that, the depth error hi and the relationship of depth-offset midpoint ( hi ) are independent of the y. For the same reason, if the measurement line is parallel to the y-axis and moves forward in the xaxis direction, then the depth estimation error hi and the relation of depth versus offset midpoint can be obtained by rotating 90˚ in space from figures 59 and 60. Figures 61-62 show that when ignoring anisotropy, the estimated depth error and the depth-offset midpoint are symmetric in shapes with respect to the vertical axis. This means that the estimated depth and the obtained relation of depth-offset midpoint are dependent on the arrangement of the measurement line. Therefore, ignoring anisotropy can distort the velocity analysis of acquired seismic signals. All three rock samples presented in Tables 11-12 show that, regardless of strong or weak, neglecting the rock anisotropy will lead to errors in the time-depth relation. For strong anisotropic rock, such as M-shale, these errors can be significant. For moderate or strong anisotropy, in practical work, the induced depth error may reach up to tens or even hundreds of meters.

6.2. VTI-TTI Media Interface In the practical world, the axis of symmetry for a sequence of rocklayers may not be parallel to the vertical axis. Instead, the sequence is a collection of tilted rock layers. This macroscopic structure of anisotropic rock is called a TTI medium. Invoking knowledge on VTI medium, the stiffness matrix of a TTI medium may be converted into a matrix which has the same format as that of a VTI medium. This can be achieved through the Bond transformation

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in rotational coordinates. Of course, an inverse transformation from the stiffness matrix of a VTI medium would yield the stiffness matrix form of a TTI medium. Let’s consider an acoustic wave at the interface between VTI and TTI media. A schematic representation of these media is shown in Figure 63. We are primarily interested in the influences of physical, geometrical, and anisotropic parameters on reflection/refraction coefficients, including incident-angle, physical and anisotropic parameters, as well as tilted angle of a TTI medium. Again, the anisotropic and mechanical properties of a VTI medium can be expressed by a stiff matrix

C (V )

 c11(V )  (V )  c12  c(V )   13  0  0   0 

c12(V ) c11(V ) c13(V ) 0 0 0

c13(V ) c13(V ) (V ) c33 0 0 0

0 0 0 (V ) c44 0 0

0 0 0 0 (V ) c55 0

        (V )  c66  0 0 0 0 0

(97)

where c12(V )  c11(V )  2c66(V ) , which means there are five independent elements.

Bond transformation with respect to a rotating angle  yield a stiff matrix for a TTI medium

C (T )  GC (V ) J 1

 c11(T )  (T )  c12  c (T )   13  0  c (T )  15  0 

c12(T ) c11(T ) c11(T ) 0 (T ) c25 0

c13(T ) c11(T ) c11(T ) 0 (T ) c35 0

0 0 0 (T ) c44 0 (V ) c46

c15(T ) (T ) c25 (T ) c35 0 (V ) c55 0

     (T ) c46  0   (V )  c66  0 0 0

(98)

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Figure 63. A schematic representation of VTI and TTI anisotropic rock models: (a) VTI model; (b) TTI model.

The two transformational matrices in Bond transformation are  cos 2   0   sin 2  G 0   (sin 2 ) / 2   0 

 cos 2    0  sin 2  J 1    0  sin 2   0 

0 sin 2  0  sin 2 0   1 0 0 0 0  0 cos 2  0 sin 2 0   0 0 cos  0 sin   0 (sin 2 ) / 2 0 cos 2 0   0 0  sin  0 cos  

(99)

0 sin 2  0 (sin 2 ) / 2 0   1 0 0 0 0  0 cos 2  0 (sin 2 ) / 2 0   0 0 cos  0 sin   0  sin 2 0 cos 2 0   0 0  sin  0 cos  

(100)

and the matrix elements are (V ) (V ) c11(T )  c11(V ) cos4   c33 sin 4   [(c13(V )  2c44 )sin2 2 ] / 2

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c12(T )  c12(V ) cos2   c13(V ) sin 2  (V ) (V ) c13(T )  c13(V ) (sin 4   cos4  )  [(c11(V )  c33 ) / 4  c44 ]sin 2 2

(V ) (V ) c15(T )  {[(c11(V )  c13(V ) )cos2   (c13(V )  c33 )sin 2  ] / 2  c44 cos 2}sin 2

(T ) c23  c12(V ) sin 2   c13(V ) cos2 

(T ) c25  [(c12(V )  c13(V ) )sin 2 ] / 2

(T ) (V ) (V ) c33  c11(V ) sin 4   c33 cos4   (c13(V ) / 2  c44 )sin 2 2

(T ) (V ) (V ) c35  {[(c11(V )  c13(V ) )sin 2   (c13(V )  c33 )cos2  ] / 2  c44 cos 2}sin 2

(T ) (V ) (V ) c44  c44 cos2   c66 sin 2 

(T ) (V ) (V ) c46  [(c66  c44 )sin 2 ] / 2

(T ) (V ) (V ) c55  [(c11(V )  c33  2c13(V ) )sin 2 2 ] / 4  c44 cos2 2

(T ) (V ) (V ) c66  c44 sin 2   c66 cos2 

The VTI-TTI media interface is in xy-plane. We set VTI medium on top of the interface and the TTI medium is the underside of the interface, as shown in Figure 64. The angle between the symmetric-axis of TTI medium and the z-axis is φ. An incident P-wave induces a set of waves, e.g., reflected P-wave, refracted P-wave, reflected SV-wave, and refracted SV-wave. Particle displacements of these waves are parallel to xz-plane, as shown in Figure 64(b). The polarization of P- and SV-waves at a given point are perpendicular to each other. The polarization directions of these two waves at different space points are usually not perpendicular.

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Figure 64. Schematic plots of VTI-TTI media interface: (a) showing the angle between the symmetric-axis of VTI and that of TTI; (b) geometry of wave front normal and polarization directions for incident wave and converted waves at the VTI-TTI media interface.

Practically, for most anisotropic media, none of the waves will be purely longitudinal nor will be purely transverse. For a harmonic vibration with an angular frequency ω, the particle displacement of the incident Pwave can be written as

 u x(0)  S (0)  R (0)  (0)  exp[i (t  k (0) x sin   k (0) z cos  )]  u   z 

(101)

The particle displacements of the four induced waves are

 u x(1)  S (1)  R (1)  (1)  exp[ j (t  k (1) x sin  (1)  k (1) z cos  (1) )] u   z 

S

(2)

 u x(2)   R  (2)  exp[ j (t  k (2) x sin  (2)  k (2) z cos  (2) ]  u   z  (2)

(102)

(103)

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 u z(3)  S (3)  R (3)  (3)  exp[ j (t  k (3) x sin  (3)  k (3) z cos  (3) )]  u   x 

(104)

 u z(4)  S (4)  R (4)  (4)  exp[ j (t  k (4) x sin  (4)  k (4) z cos  (4) )] u   x 

(105)

The subscripts {i} = {x, y, z} denote Cartesian coordinates; the superscripts {m} = {0, 1, 2, 3, 4} denote specific wave, e.g., incident P-wave (m = 0), reflected P-wave (m = 1), refracted P-wave (m = 2), reflected SV-wave (m = 3), and refracted SV-wave (m = 4);  (m ) is an incidence/reflection/refraction angle of corresponding velocity; R( m) | R(m) | exp[i (m) ] and  ( m)  atan  Im R( m) / Re R( m)  ;R(1) is the reflection

coefficient of P-wave to P-wave; R(2)is the refraction coefficient of P-wave to P-wave; R(3)is the reflection coefficient of P-wave to SV-wave, R(4) is the refraction coefficient of P-wave to SV-wave; R(0)=1. From equations (101)-(105), it is possible to derive an analytical expression to solve for polarization coefficients and phase velocities, which leads to an eighth-order polynomial of sine-function of the refraction angles for the reflected P- and SV-waves. An efficient algorithm can be developed in calculations of reflection/refraction coefficients at VTI-TTI media interface.

6.2.1. Phase Velocity Solutions In absence of body forces, Kelvin-Christoffel equation can be simplified for the incident P-wave and the reflected/refracted P- and SVwaves induced at VTI-TTI media interface  ( m )   ( m ) 2  11  (m) 31 

 ( m)   ux   0 2   ( m)  (m) 33   ( m )    u z  ( m) 13

(106)

For the incidence wave from VTI medium, the matrix elements are

94

Lin Fa and Meishan Zhao (0,1,3) (V ) 11  A11(V ) sin 2  (0,1,3)  A44 cos2  (0,1,3)

(107)

(0,1,3) (0,1,3) (V ) 13  31  [ A13(V )  A44 ]sin  (0,1,3) cos (0,1,3)

(108)

(0,1,3) (V ) 33  A33(V ) cos2  (0,1,3)  A44 sin 2  (0,1,3)

(109)

The phase velocity solutions of incident P-wave and reflected P- and SV-waves are (0,1,3) v1,2  {[ A4(V ) sin 2  (0,1,3)  A5(V )  Q( (0,1,3) )] / 2}1/2

(110)

Q( (0,1,3) )  [( A1(V ) sin 2  (0,1,3)  A2(V ) cos2  (0,1,3) )2  ( A3(V ) sin 2 (0,1,3) )2 ]1/ 2 (111)

where (V ) , A2(V )  A44(V )  A33(V ) , A3(V )  A13(V )  A44(V ) , A1(V )  A11(V )  A44

(V ) , Aij  Cij /  A4(V )  A11(V )  A33(V ) , A5(V )  A33(V )  A44

(V )

(V )

(V )

In equations (110)-(111), the superscript V denotes the VTI medium.

 (V ) is the density of VTI medium. The subscripts 1 and 2 indicate that there are two different phase velocity solutions. For the refraction medium TTI, the matrix elements in KelvinChristoffel equation are ( 2, 4) (T ) (T ) (T ) 11  A11 sin 2  (2,4)  A15 sin 2 ( 2,4)  A55 cos2  ( 2,4)

(112)

(T ) (T ) (T ) 13( 2, 4)  31( 2, 4)  ( A13(T )  A55 ) sin 2 ( 2, 4) / 2  A35 cos2  ( 2, 4)  A15 sin 2  ( 2, 4)

(113) ( 2, 4 ) (T ) (T ) (T ) 33  A33 cos2  ( 2, 4)  A35 sin 2 ( 2, 4)  A55 sin 2  ( 2, 4)

(114)

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and the phase velocity solutions of refracted P- and SV-waves are v1(,22, 4)  {[c1 sin 2  ( 2, 4)  c 2 sin 2 ( 2, 4)  c3 cos2  ( 2, 4)  Q( ( 2, 4) )] / 2]}1 / 2 (115) Q (2,4)  [b1 sin 4  (2,4)  b2 cos 4  (2,4)  b3 sin 2 

(116)  sin 2 (2,4) (b4 sin 2  (2,4)  b5 cos 2  (2,4) )]1/2

where,  (T ) is the density of TTI medium, and a1  A11(T)  A55(T) , a 2  A15(T)  A35(T) , a3  A55(T)  A33(T) , a 4  ( A13(T)  A55(T) ) / 2 , b1  a12  4[ A15(T) ] 2 , b2  a32  4[ A35(T) ] 2 , b3  a1 a3 / 2  a 22  4a 42  2 A15(T) A35(T) , b4  2(a1 a 2  4a 4 A15(T) ) , b5  2(a2 a3  4a4 A35(T) ) ,

c1  A11(T)  A55(T) , c 2  A15(T)  A35(T) , c3  A33(T)  A55(T) , A (jiT )  c (jiT ) /  (T ) .

6.2.2. Polarization Coefficients Propagating on xz-plane, the polarization coefficients of the incident and the induced waves satisfy ( m) 11  (v1(,m2 ) ) 2 u z( m)   ( m) u x( m) 13

u x( m ) u z( m )



( m) 33  (v1(,m2 ) ) 2 ( m) 13

( m) ( m) * ( m) ( m) * Under the normalization u x [u x ]  u z [u z ]  1 we have

(117)

(118)

96

Lin Fa and Meishan Zhao (u x( m) )* u z( m)  

u x( m) (u z( m) )*  

( m) ( m) * 13 (13 ) ( m) * ( m) ( m) ( m) (13 ) [33  (v1(,m2 ) ) 2 ]  13 [11  (v1(,m2 ) ) 2 ]*

( nm ( m) * 13 (13 ) ( m) ( m) ( m) * ( m) 13 [33  (v1(,m2 ) ) 2 ]*  (13 ) [11  (v1(,m2 ) ) 2 ]

(119)

(120)

where the superscription * signifies the complex conjugate. In pre-critical angle area, all incident and induced waves are homogeneous with the following polarization coefficients 1/ 2

u

( m) x

   ( m )  (v ( m ) ) 2   ( m) 33 ( m) 1,2 ( m) 2       2(v )  33 1,2  11 

u

( m) z

( m) ( m) 2   11  (v1,2 )   ( m)    ( m)  2(v( m) )2  33 1,2  11 

(121)

1/ 2

(122)

In pos-critical angle region, the polarization coefficients of induced homogeneous waves are u x( m)  (cos ( m)  i sin  ( m) ) 

(n) ( n) * 13 [13 ] (n) ( n) * 13 [13 ]

(n)  {11

(123)

(n)  [v1(,m2 ) ]2 }{11

 [v1(,n2) ]2 }*

( m) ( m) * 13 [13 ] ( m) ( m) 2 ( m)  {33  [v1,2 ] }{33

 [v1(,m2 ) ]2 }*

u z( m)  (cos ( m)  i sin  ( m) ) 

( m) ( m) * 13 [13 ]

(124)

In the pos-critical angle region, the polarization coefficients of induced inhomogeneous waves for left-rotational elliptical-polarization are

Recent Progress in Acoustical Theory and Applications

u x( m)

u



i

(m) z

( m) ( m) 13 [33  (v1(,m2 ) ) 2 ] ( m) * ( m) ( m) ( m) (13 ) [33  (v1(,m2 ) ) 2 ]  13 [11  (v1(,m2 ) ) 2 ]

(m) 13( m ) (33  (v1(,m2 ) ) 2 ) (m) (13( m ) )* (33  (v1(,m2 ) ) 2 )  13( m ) (11( m )  (v1(,m2 ) ) 2 )

97

(125)

(126)

and the polarization coefficients for right-rational elliptical-polarization are

u

( m) x

u

i

(m) z



( m) 13( m ) [33  (v1(,m2 ) ) 2 ] ( m) (13( m ) )*[33  (v1(,m2 ) ) 2 ]  13( m ) [11( m )  (v1(,m2 ) ) 2 ]

(m) 13( m ) [33  (v1(,m2 ) ) 2 ] (m) (13( m ) )*[33  (v1(,m2 ) ) 2 ]  13( m ) [11( m )  (v1(,m2 ) ) 2 ]

(127)

(128)

6.2.3. High Order Polynomials for Reflection/Refraction Angle For P-wave incident to VTI-TTI media interface with incident-angle θ from the VTI medium, Snell’s law provides two equivalent relationships ( m) constV ( )   sin  ( m ) / v1,2 

2

constV ( )   sin  ( m ) / v1(0)   2sin 2  /  A4(V ) sin 2   A5(V )  Q( )  2

(129)

(130)

Within the incidence VTI medium, in combination of above equation with phase velocity equation yields to a 4th polynomial for solving reflection angles of reflected P- and SV-waves

98

Lin Fa and Meishan Zhao B1(1,3) ( )sin 4  (1,3)  B3(1,3) ( )sin 2  (1,3)  B5(1,3) ( )  0

(131)

where B1(1,3) ( )  A1(V )  2[ A1(V ) A2(V )  2( A3(V ) )2 ]  A2(V )  [ K (V ) ( )]2 B3(1,3) ( )  2[ A1(V ) A2(V )  ( A3(V ) )2  ( A2(V ) )2  A5(V ) K (V ) ( )] B5(1,3) ( )  A2(V )  ( A5(V ) )2 , K (V ) ( )  2 / constV ( )  A4(V )

Within TTI medium for refraction, it yields an 8th polynomial for solving refraction angles of refracted P- and SV-waves f1(2,4) ( )sin 8  (2,4)  f 2(2,4) ( )sin 6  (2,4)  f3(2,4) ( )sin 4  (2,4)  f 4(2,4) ( )sin 2  (2,4)  f5(2,4) ( )  0

(132)

where

f1(2,4) ( )  (d12  d 22 )2  8d1d3  8d 2 d3  16d32  4d 42  8d 4 d5  d52 f 2(2,4) ( )  4(d1d 2  d 22  2d1d3  6d 2 d3  8d32  d 42  4d 4 d5  3d52 ) f3(2,4) ( )  2(d1d 2  3d 22  12d 2 d3  8d32  4d 4 d5  6d52 )

f 4(2,4) ( )  4(d22  2d 2 d3  d52 ) ,

f5(2,4) ( )  d 2 ,

c4  constV ( ) / 2  c1

d1  b1  c42 , d 2  b2  c32 , d3  b3  c22  c3c4 / 2 , d4  b4  2c2c4 , d5  b5  2c2c3

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6.2.4. The Matrix Equation of Reflection/Refraction Coefficients A matrix equation for particle displacement versus stress can be expressed as T (m)  C (n)： s S (m)

(133)

where, T(m) is stress tensor, C(n) is the stiffness matrix, ▽S is the gradient operator, “:” is tenser product operator, and the superscripts {n}={V, T} stand for VTI medium and TTI medium respectively. With boundary conditions for continuity of normal stress components, tangential, and normal components of particle displacement, a matrix equation for reflection/refraction coefficients as a function of incidentangle, polarization coefficients, and physical parameters of the media can be obtained  M 11 M 12   M 21 M 22  M 31 M 32   M 41 M 42

M 13 M 23 M 33 M 43

M 14   R (1)   N1      M 24   R (2)   N 2   M 34   R (3)   N 3      M 44  R (4)   N 4 

where, ( v ) ( 0) N1  k (0) (c13(v) u x(0) sin (0)  c33 u z cos (0) )

(v) N 2  k (0) c44 (u x(0) cos (0)  u z(0) sin (0) )

N 3  u x(0) , N 4  u z(0) , ( v ) (1) M11  k (1) (c13(v)u x(1) sin (1)  c33 u z cos (1) )

( 0) (3) M12  k (3) (c13(v)u z(3) sin (3)  c33 u x cos (3) )

(134)

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Lin Fa and Meishan Zhao

M13  k (2) (c13(T )ux(2) sin (2)  c33(T )uz(2) cos (2) )  k (2) c35(T ) (ux(2) cos (2)  uz(2) sin (2) ) M14  k (4) (c13(T )uz(4) sin (4)  c33(T )ux(4) cos (4) )  k (4) c35(T ) (uz(4) cos (4)  ux(4) sin (4) )

M 21  k (1) c44(0) (ux(1) cos (1)  uz(1) sin (1) ) , M 22  k (3)c44(0) (u z(3) cos (3)  u x(3) sin (3) ) M 23  k (2) (c15(T )u x(2) sin (2)  c35(T )u z(2) cos (2) )  k (2) c55(T ) (u x(2) cos (2)  u z(2) sin (2) )

M 24  k (4) (c15(T )u z(4) sin (4)  c35(T )u x(4) cos (4) )  k (4) c55(T ) (u z(4) cos (4)  u x(4) sin (4) )

M 31  u x(1) , M 32  uz(3) , M 33  ux(2) , M 34  uz(4) , M 41  uz(1) , M 42  ux(3) M 43  u z(2) , M 44  ux(4) . At the VTI-TTI media interface, the coefficients ( 2, 4 ) ( 2, 4 ) f1( 2, 4) ( ) , f 2 ( ) , f 3 ( )

B1(1,3) ( )

, B3(1,3) ( ) ,

and f 4(2,4) ( ) are functions of the incident angle  ;

whereas the coefficients B5(1,3) and f 5( 2,4) are not relative to the incident angle  . Let’s consider two model systems for numerical calculations with parameters shown ion Table 13. System 1 is an interface of T-sandstone (as an incidence medium) and A-shale (as a refraction medium). System 2 is an interface of MDMI and MDMR. MDMI stands for modeling incidence medium and MDMR stands for modeling refraction medium. The anisotropic parameters of a rock are related to the rock stiffness matrix elements c13   [ * 4  ( 2   2 )[(  1) 2   2 ]   2 , c33   2 , c11  (2  1)  2 ,

c44   2 , c66  (2  1)  , where,  and  are the phase velocities of P-and SV-waves in symmetric axis direction of the anisotropic rocks (VTI medium).

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Table 13. Anisotropic parameters for T-sandstone (Taylor sandstone), A-shale (anisotropic shale), MDMI (model incidence medium), and MDMR (model refraction medium) Sample

 (m/s)

 (m/s)



*



 (g/cm3 )

T-sandstone A-shale MDMI MDMR

3368 2745 2310 3060

1829 1508 1330 1770

0.1100 0.1030 0.2213 0.2213

-0.1270 -0.0730 0.0000 0.0000

0.2550 0.3450 0.0000 0.0000

2.5000 2.3400 2.0400 2.2100

Figure 65. The calculated slowness for two models: (a) model 1, an interface of Tsandstone and A-shale; (b) model 2, an interface of MDMI and MDMR. The upside of the interface is a VTI (incidence) medium and the downside is a TTI medium. Thin lines are the slowness of the P-wave and thick curves are that of SV-wave.

The slowness curves for models 1-2 are calculated and are shown in Figure 65. System 1 does not have critical incident-angle. Figure 65(a) shows that the slowness of P-wave in the incidence T-sandstone medium is smaller than the slowness of P- and SV-waves in the refraction A-shale medium in all directions. Figure 65(b) shows that the slowness of P-wave in incidence medium is greater than that of P-wave and smaller than that of SV-wave in the refraction medium. So, System 2 is an example where there is a critical incident-angle corresponding to the refracted P-wave.

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Lin Fa and Meishan Zhao

Figure 66. The reflection/refraction coefficients of System 1 vs. incident angle θ at different values of ε(1): (a) the modulus and phase of R(1); (b) those of R(2) (c) those of R(3); (d) those of R(4).

The anisotropy of the media could lead to modulation on the amplitude of reflection/refraction coefficients and its corresponding phase in post critical-angle region; and the same effect would come from the tilt-angle of the refraction medium. However, the effect of tilt-angle on the reflection/refraction coefficients is greater than that of anisotropic parameters. Specifically, the effects of ε(1) on R(1), R(2), R(3)and R(4) are presented in Figure 66. To satisfy the law of energy conservation, the z-component of power density flux of incident P-wave must be equal to the summation of zcomponent real-parts of energy density flux, including all converted waves induced on the VTI-TTI media interface. Because the converted waves induced at VTI-TTI media interface are homogeneous, the z-components of their energy flux density are real. The real parts of z-components are given in Figure 67. The anisotropy of the

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103

incidence medium affects the magnitude of power flux density of incident P-wave in z-direction; but the anisotropy of the refraction medium has no effect on power flux density. The anisotropy of the refraction medium can influence the magnitude of power density flux in z-direction on the real-part of each induced wave. It has no effect on the total power density flux in z-direction with respect to the real-parts of all converted waves. For System 2 of VTI-TTI interface, the derivative of a reflection/refraction coefficient with respect to the incident angle is not continuous. For post-critical-angle incidence, the reflection/refraction coefficients become complex numbers, which can leads to the induced waves to create a lateral-displacement/transition-time relative to incident P-wave on the interface.

Figure 67. The values of |Re{ Pz(m ) }| vs. incident angle for different values of ε(1) at φ=30°.

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Lin Fa and Meishan Zhao

Similar to System 1, the reflection/refraction coefficients from the influence of ε(1) are shown in Figure 68. The results for power flux densities in z-direction are presented in Figure 69.

Figure 68. The reflection/refraction coefficients of System 2 vs. θ for several different values of ε(1): (a) the modulus and phase of R(1); (b) those of R(2) (c) those of R(3); (d) those of R(4).

Figure 69.The values of |Re{ Pz(m ) }| vs. incident angle for different values of ε(1) at φ=30°, where blue, red and black curves stand for the cases of ε(1)=0.0, 0.103 and 0.2132.

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CONCLUSION We have reviewed the recent progress in acoustical theory and applications, focusing on the development in our group [14, 40, 49, 51-63]. There are several aspects of acoustic theories and applications that we have paid special attention to, including Goos-Hänchen effect in acoustics, anomalous incident angles from scattering on an interface, anisotropic effects on acoustic wave propagation, fluid-solid interface, and solid-solid interface. On industrial applications, we focused on acoustic logging and seismic data analysis. We have also discussed some modeling of measurement network and presented some interesting results on acoustic/electric and electric/acoustic conversions. Furthermore, we have put great efforts in identifying the significance of physical and geometric factors on reflection/refraction coefficients and polarization states in systems of interfaces (both fluid-solid interface and solid-solid interface), e.g., anisotropy in media, materials used for a measurement system, geometric structure of the measurement systems, acoustic signals in excitation, and incidence angles of the wave propagation. The predicted results from our numerical calculations are generally in good agreement with experimental data. It is important to point out that there are other numerous existing studies on modeling the measurement networks, which are not discussed in this chapter but are of general interest to acoustic measurement theory. Moreover, we have derived an eighth-order polynomial with regard to the refraction angle for refracted P- and SV-waves induced at VTI-TTI media interface. Accordingly, we have proposed an efficient algorithm in solving this eighth-order polynomial for reflection/refraction coefficients. The theoretical derivation is checked from the law of energy conservation through numerical calculations. The studies on reflection coefficient at VTI-TTI media interface provides insightful information into the analysis of amplitude variations with offset (AVO) for exploring accurate the new oil reservoirs or gas fields.

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We now comment on the difference between the conventional and the improved measurement networks. In this aspect, we have discussed our proposed and the improved network models for acoustical application to petroleum logging and seismic exploration. It is critical to identify the intrinsic relationships amongst various physical quantities, e.g., drivingvoltage signal, electric-acoustic conversion of source-transducers, acousticelectric conversion of receiver-transducers, measured logging signals, and the propagation media. The newly proposed network model describes the acoustic-logging process-based strictly on a physical mechanism. Amongst the most important physical concepts that we have proposed are acousticelectric analog and driving voltage convolution with electric-acoustic impulse response. The latter concept has led to practical and improved petroleum logging network. It is practically much closer to the actual situation of acoustic logging than any earlier acoustic-logging models. Applications of the new network model in analysis of acoustic-logging process lead to accurate acoustic-logging information, such as propagation velocity, signal amplitude, wavelet phase, and frequency spectrum of acoustic-logging signal in the formation. There are several open questions to be answered. For example, from the studies of a fluid-solid interface (water–Perspex interface) in the postcritical-angle incidence region, the analysis suggests that there is indeed a Goos–Hänchen effect in acoustics. This theoretical advance provides us a confidence in applying some optical theory to acoustics. However, it is interesting to note that a Goos–Hänchen displacement in optics is a coherent effect of the total reflection of a finite-sized optical beam with a harmonic frequency. Whereas, the transition of an acoustic signal is a reflection incoherent interference effect of non-total reflection of different frequency components. Furthermore, from our analysis of acoustic Goos– Hänchen analog, we must conclude that the observed first arrival of the acoustic signal is a reflected P-wave, not the sliding refracted P-wave as conventionally specified in the traditional acoustic-logging model. This is still and indeed an intriguing issue that requires further investigation.

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ACKNOWLEDGMENTS This work is supported in part by Xi’an University of Posts and Telecommunications, National Natural Science Foundation of China (Grant no. 41974130), and the Physical Sciences Division at The University of Chicago.

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[2]

[3]

[4] [5] [6] [7] [8]

[9]

Grady, N., Heyes, J., Chowdhury, D. et al. (2013). “Metamaterials for Broadband Linear Polarization Conversion and Near-Perfect Anomalous Refraction.” Science. 340, 1304: 1-29. Yu, N. F., Genevet, P., Kats, M. A. et al. (2011). “ Light Propagation with Phase Discontinuities: Generalized Laws of Reflection and Refraction.” Science. 334, 6054: 333-337. Fa, L., Xue, L., Fa, Y. X. et al. (2017). “Acoustic Goos-Hänchen effect.” Science China: Physics, Mechanics & Astronomy 60, 10: 104311. Thomsen, L. (1986). “Weak anisotropy.” Geophys. 51, 10: 19541966. Wang, Z. (2002). “Seismic anisotropy in sedimentary rocks, part 1: A single-plug laboratory method.” Geophys. 67, 5: 1414-1422. Wang, Z. (2002). “Seismic anisotropy in sedimentary rocks, part 2: A single-plug laboratory method.” Geophys. 67, 5: 1423-1440. Daley, P., Hron, F. (1977). “ Reflection and transmission coefficients for transversely isotropic media.” Bull., Seis. Sot. Am. 67:661-675. Daley, F., Hron, F. (1979). Reflection and transmission coefficients for seismic waves in ellipsoidally isotropic media.” Geophys. 44: 2738. Tsvankin, I. “Seismic signature and analysis of reflection data in anisotropic media.” Elsevier, 2005.

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[10] Carcione, J. “Wave fields in real media: wave propagation in anisotropic, anelastic and porous media.” Elsevier, 2001. [11] Klimeš, L. (2003). “Weak-contrast reflection-transmission coefficients in a generally anisotropic background.” Geophys. 68: 2063-2071. [12] Fa, L., Brown, R., Castagna J. (2006). “Anomalous postcritical refraction behavior for certain transversely isotropic media.” J. Acoust. Soc. Am.120: 3479-3492. [13] Fa, L., J. P. Castagna, J., Dong, H. (2008). “An accurately fast algorithm of calculating reflection/transmission coefficients.” Sci. China-Phys. Mech. Astron. 51: 823-846. [14] Fa, L., Tang, J., Zhao, M. et al. (2019). “Reflection/refraction of acoustic wave on VTI-TTI media interface.” Frontiers of Physics. [15] Goos, F., Hänchen, H. (1947). “Ein neuer und fundamentaler versuch zur Totalreflexion.”Ann. Phys. 436: 333-346. [16] Lotsch, H. K. V., H. K. V. (1970). “Beam displacement at total reection: the Goos-H¨ nchen effect.” Optik 32:116-137, 189-204, 299-319, 553-569. [17] Wild, W. J., Giles, C. L. (1982). “Goos-hänchen shifts from absorbing media.”Phys. Rev. A. 25, 4: 2099-2101. [18] Negative goos–hänchen shift at metal surfaces.” Opt. Commun. 276, 2: 206-208. [19] Lai, H. M., Chan, S. W. (2002). “Large and negative Goos–Hänchen shift near the Brewsterdip on reflection from weakly absorbing media.” Opt. Lett. 27, 9: 680-682. [20] Tamir, T. (1986). “Nonspecular phenomena in beam fields reflected by multilayered media.” J. Opt. Soc. Am. A.3, 4: 558-565. [21] Nasalski, W. (1989). “Modified reflectance and geometrical deformations of gaussian beamsreflected at a dielectric interface.” J. Opt. Soc. Am. A.6, 9: 1447-1454. [22] Tournois, P. (1997). “Negative group delay times in frustrated Gires.”IEEE J. Quantum Electron. 33: 519-526.

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[23] Ranfagni, A., Fabeni, P., G. P. Pazzi, Mugnai, D. (1993). “Anomalous pulse delay in microwave propagation: A plausible connection to the tunneling time.” Phys. Rev. E 48:1453. [24] Lakhtakia, A. (2003). “On Planewave Remittances and Goos– Hänchen Shifts of Planar Slabs with Negative Real Permittivity and Permeability.”Electromagnetics 2: 71-75. [25] Resch, K. J., Lundeen, J. S., Steinberg, A. M. IEEE J. (2001). “Total reflection cannot occur with a negative delay time.” QuantumElectron. 37: 794-799. [26] Breazeale, M. A., Adler, L., Scott, G. W. (1977). “Interaction of ultrasonic waves incident at the Rayleigh angle onto a liquid-solid interface.” J. Appl. Phys. 48: 530.-537. [27] Atalar, A., Quate, C. F., Wickramasinghe, H. K. (1977). “Phase imaging in reflection with the acoustic microscope.” Appl. Phys. Lett.31: 791. [28] Briers, R., Leroy, O., Shkerdin, G. (2000). “Bounded beam interaction with thin inclusions. Characterization by phase differences at Rayleigh angle incidence.” J. Acoust. Soc. Am. 108: 1622-1630. [29] Wang, A. L., Liu, F. P. (2014). “The lateral shift of reflection Shwave on an interface of two Media.” Appl. Mech. Mater. 488-489: 923-925. [30] Liu, F. P., Wang, A. L., Li, R. Z., Chen, H. G., Yang, C. C. (2009). “The influence on normal moveout of total reflected SH-wave by Goos Hänchen effect at an interface of strata.” Chin. J. Geophys. 52, 8: 2128-2134. [31] Liu, F. P., Meng, X. J., Xiao, J. Q., Wang, A. L., Yang, C. C. (2012). “The Goos-Hänchen shift of wide-angle seismic reflection wave.” Sci. China Earth Sci. 55, 5: 852-857. [32] Ricker, N. (1953). “Wavelet contraction, wavelet expansion, and the control of seismic resolution.” Geophys. 18, 4: 769. [33] Tsang, L., Rader, D. (1979). “Numerical evaluation of the transient acoustic waveform due to a point source in a fluid-filled borehole.” Geophys. 44, 10: 1706.

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[34] Tsang, L., Rader, D., Cheng, C. H., Toksöz, M. N. (1981). “Elastic wave-propagation in fluid-filled borehole and synthetic acoustic logs.” Geophys. 46, 7: 1042-1053. [35] Kurkjian, A. L. (1985). “Numerical computation of individual farfield arrivals excited by an acoustic source in a borehole.” Geophys. 50, 5: 852. [36] Gong, W. “Full waveform analysis of acoustic well logging and its multidimensional Procession Methods.” Dissertation for the Doctoral Degree, Sourtheast University, Najing, 1988. [37] Fa, L., Ma, H. (1991). “Design of a new type of array transmitting sonic sonde.”Acta Petrol. Sin. 12: 52-57. [38] Gibson R. L., Peng, C. (1994). “Low- and high-frequency radiation fromseismic sources in cased boreholes.”Geophys. 59, 11: 1780. [39] Fa, L., Castagna, J. P., Suarez-Rivera, R., Sun, P. (2003). “An acoustic-logging transmission-network model (continued): Addition and multiplication ALTNs.” J. Acoust. Soc. Am. 113, 5: 2698-2703. [40] Fa, L., Xie, W. Y., Tian, Y., Zhao, M. S., Ma, L., Dong, D. Q. (2012). “Effects of electric-acoustic and acoustic-electric conversions of transducers on acoustic logging signal.” Chin. Sci. Bull. 57, 11: 1246-1260. [41] Backus G. L. (1962). “Long-wave elastic anisotropy produced by horizontal layering.” J. Geophys. Res.67: 4427–4440. [42] Ĉervenŷ, V. “Seismic Ray Theory.” Cambridge University Press, 2001. [43] Tsvankin, I. (1996). “P-wave signatures and notation for transversely isotropicmedia: An overview.” Geophys. 61: 467–483. [44] Kim, K. Y., Wrolstad, K. H., Aminzadeh. F. (1993). “Effects of transverse isotropy on P-wave AVO for gas sands.” Geophys. 58: 883–888. [45] Rüger, A. (1997). “P-wave reflection coefficients for transversely isotropic models with vertical and horizontal axis of symmetry.” Geophys. 62: 713–722. [46] Banik, N. C. (1984). “Velocity anisotropy of shales and depth estimation in the North Sea Basin.” Geophys. 52: 1654–1664.

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[47] Sams, M. S., Worthington, M. H., Khanshir M S. (1993). “A comparison of laboratory and field measurements of P-wave anisotropy.” Geophys. Prospect. 41: 189–206. [48] Auld, B. A.“Acoustic fields and waves in solids.” Wiley, 1972. [49] Fa, L., Zhao, M.,Liu, Y. C., Wang, L., Wang, Y. Q., and Sun, J. G. (2014). “Polarization of plane wave propagating inside elastic hexagonal system solids.” Sci. China-Phys. Mech. Astron. 57: 1-12. [50] Ostrander, W. J., W. J. (1984). “Plane-wave reflection coefficients for gas sands at nonnormal angles of incidence.”Geophys. 1984, 49, 10: 1637-1648. [51] Fa, L., Li, W. Y., Zhao, J., Han, Y. L., Liang, M., Ding, P. F., and Zhao, M. S. (2017). “Polarization state of an inhomogenously refracted compressional-wave induced at interface between two anisotropic rocks.” J. Acoust. Soc. Am. 141: 1-6. [52] Fa, L., Zhao, J., Han, Y. L., Li, G. H., Ding, P. F., and Zhao, M. S. (2016). “The influence of rock anisotropy on elliptical-polarization state of inhomogenously refracted P-wave.” Sci. China-Phys. Mech. Astron. 59: 644301 [53] Fa, L., Xue, L., Fa, Y. X., Han, Y. L., Zhang, Y. D., Cheng, H. S., Ding, P. F., Li, G. H., Bai, C. L., Xi, B. J., Zhang, X. L., and Zhao, M. S. (2017). “The acoustic Goods-Hanchen effect.” Sci. ChinaPhys. Mech. Astron. 60: 104311. [54] Fa, L., Mou, J. P., Fa, Y. X., Zhou, X., Zhang, Y. D., Liang, M., Wang, M. M., Zhang, Q., Ding, P. F., Feng, W. T., Yang, H., Zhao, M. S. (2018). “On transient response of piezoelectric transducers.” Frontiers in Physics. 6: 123. [55] Fa, L., Zhou, X., Fa, Y. X., Zhang, Y. D., Mou, J. P., Liang, M., Wang, M. M., Zhang, Q., Ding, P. F., Feng, W. T., Yang, H., Zhao, M. S. (2018). “An innovative model for the transient response of a spherical thin-shell transducer and an experimental confirmation.” Sci. China-Phys. Mech. Astron. 61: 104311. [56] Fa, L., Zeng, Z., Deng, C., and Zhao, M. (2010). “Effects of Geometrical-Size of Cylindrical-shell Transducer on Acoustic-beam

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[58]

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Lin Fa and Meishan Zhao Steering Efficiency for a Slim-hole Acoustic-logging Tool.” The Open Acoustics Journal 3: 21-29. Fa, L., Castagna, J. P., Zeng, Z., Brown, R. L., and Zhao, M. (2010). “Effects of anisotropy on Time-Depth Relation in Transversely Isotropic Medium with a Vertical Axis of Symmetry,” Chin. Sci. Bulle. 55: 2241. Zhao, Y., Zhao, N., Fa, L., and Zhao, M. (2013). “Seismic Signal and Data Analysis of Rock Media with Vertical Anisotropy,” Journal of Modern Physics 4: 11-18. Fa, L., Tian, Y., Xie, W., and Zhao, M. (2012). “Effects of ElectricAcoustic and Acoustic-Electric Conversions of Transducers on Acoustic-Logging Signal”, Chin. Sci. Bulle. 57: 1246-1260. Fa, L., Wang, L., Liu, L., Zheng, Y., Zhao, Y., Zhao, N., Li G., and Zhao, M. (2013). “Research Progress in Acoustical Application to Petroleum Logging and Seismic Exploration,” The Open Acoustics Journal 6: 1-10. Fa, L., Qi, Z., Wenhui, C., Ding, P. F., Liang, M., Mou, J. P., Tang, J. J., Zhang, Y., Zhao, M. (2019). “Theoretical study and experimental Verification of a transient response model of thin spherical shell piezoelectric-transducer,” Journal of Vibration Engineering xx: xx. Fa, L., Fa, Y., Zhang, Y. D., Ding, P. F., Gong, J., Li, G. H., Li, L., Tang, S. J., and Zhao, M. (2015). “Anomalous incident-angle and elliptical-polarization rotation of an elastically refracted P-wave,” Scientific Reports 5: 12700. Fa, L., and Zhao, M. “Recent development of an acoustic measurement system.” in Understanding Plane Waves. xxx: Nova Science Publishers, 2019 (companion chapter in this book).

In: Understanding Plane Waves Editor: William A. Cooper

ISBN: 978-1-53616-779-5 © 2020 Nova Science Publishers, Inc.

Chapter 2

RECENT DEVELOPMENT OF AN ACOUSTIC MEASUREMENT SYSTEM Lin Fa1 and Meishan Zhao2,* 1

School of Electronic Engineering, Xi’an University of Posts and Telecommunications, Xi’an, Shaanxi, China 2 James Franck Institute and Department of Chemistry, University of Chicago, Chicago, IL, US

ABSTRACT We discuss a newly developed acoustic-measurement system with emphasis on measurement process and recent improvements that make an acoustic-measurement more accurate. This system is based on a model electric-acoustic transmission-network which consists of a series of parallel-connected equivalent-circuits. For acoustic-transducers, we place special emphasis on the importance of the contribution from each individual frequency component of an excitation signal, the propagation medium, and the cumulative signal-output from transducer’s mechanic/electric terminals. The system has been tested for realistic *

Corresponding Author’s Email: [email protected].

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Lin Fa and Meishan Zhao acoustic transmission. Based on the results of the measurement, we conclude that this system is easy to operate, user friendly, and highly accurate.

Keywords: acoustic measurement, acoustic network, transducers, gated sine-wave, transmission, piezoelectric material PZT4, plane wave refection/refraction, convolution, linear superposition

INTRODUCTION Acoustical measurements are extensive and far-reaching. Practical applications are ubiquitous and imperative, from geophysical sciences, biological sciences, to 5G mobile and internet communications [1-4]. The widespread geophysical applications include exploration of underground mineral resources, e.g., oil, gas, coal, metal ores, etc. [5], the measurement of the in-situ stresses of underground rock formation [6], early warning system of the dam damages and natural hazards, and more [7-13]. Applications in biological sciences are similarly extensive, including intravascular ultrasound [14], medical imaging [15], biometric recognition [16], implantable micro-devices [17], etc. The applications of acoustical measurements in mobile and telecommunications are obvious and profound in our daily life [18-19]. The quality of communications and technologies are directly touching on our life and social activities, e.g., rangefinders [20], and nondestructive detection, etc. [21-23]. Specific instrumentations must be developed for various practical acoustic applications, e.g., experimental verification of acoustic lateraldisplacement [24], inspection of a specific polarization state of a plane wave propagating inside layered isotropic/anisotropic media [25-26], to mention a few. Many times, an acoustic measurement process includes one or more conversions of electric-acoustic/acoustic-electric signals through transducers. The effect of conversions can be significant on the transmitted signal. Nevertheless, in most of the published literatures, there have not been adequate attention given to the influence of these conversions on the

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transmitted wave. Now, the questions are what the influences of these conversions are on the transmitted signal and how to make corrections. Do we lose signal during the conversions? Is there any signal distortion from a conversion? With potential issues are from signal conversions and how can we develop good instrumentations to correct them? In this chapter, we describe an acoustic-measurement system based on an acoustic-measurement transmission network [27-28]. This system has a theoretical base strictly following a physical mechanism. From this system, one can achieve accurate analysis and inversion-interpretation for the measured acoustic-signal. The relationships amongst various physical quantities can be analyzed to achieve an enhanced understanding of the signal transmission. To mention a few, these factors include the driving electric-signal wavelet, the electric-acoustic/acoustic-electric conversion physical factors, the propagation media, and the measured acoustic signal. The acoustic-measurement transmission network is a parallelly connected circuits-network which has a theoretically foundation from the principle of linear superposition based on Fourier transform of an acoustic-signal wavelet. Technically, in developing such an instrument to perform data acquisition, we employed a high resolution, high sampling rate digitizer, specifically, a sampling rate range of 500 KS/s to 15000 KS/s, with a resolution range from 16-bit to 24-bit. With such a digitizer, the system can achieve accurate measurement for a large frequency range, even for weak acoustic-signals. From acquired data, one may inspect the properties of the transducers, obtain the physical parameters of the measured fluid/solid materials, check the quality of the measured objects, and perform verification-check of existing/on-going scientific research. As an example, we will discuss the piezoelectric spherical thin-shell transducer’s transient response to test the apparatus. This application provides a practical test on the quality of the system with respect to its technical functionality. With a graphic human-machine interface, the system is easy to operate, user friendly, and highly accurate in measurement. The measured results are in good agreement with that of theoretical predictions. This newly reported acoustical-measurement

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system should be useful for both scientific research and industrial application.

THEORETICAL MODELING Spherical Shell Transducers Consider a piezoelectric spherical thin-shell transducer polarized in the radial direction. Its electrodes are connected to the inner and the outer surfaces of the shell, as shown in Figure 1. We consider the average radius of this transducer to be rb  (r0  r1 )/2 . If we select a thin-shell such that the thickness of this spherical shell is much smaller than the radius of the transducer, i.e., lt  (r1  r0 )  r0 , then we may carry out the analysis with

r0  r1  rb for a reasonably good approximation. For a piezoelectric transducer with a harmonic vibrational motion, the equations of motion can be solved to establish its corresponding circuits [5, 29-30], as shown in Figure 2. It should be noted that there are two possible equivalent circuits which are assembled from different electric/mechanic components. As shown in Figure 2, the equivalent circuits can be excited by a harmonic electric/acoustic signal to produce an objective response, where Figure 2(a) is for a source-transducer and Figure 2(b) is for a receiver-transducer. U1(t) is a power-driving voltage source. Ro is the output resistance of the power-driving voltage. Ri is the input resistance of the measurement circuit. V(t) is the power-voltage which is the signal at electric-terminals of the source. U3(t) is the electric-signal at electricterminals of the receiver. vr(t) is the vibration velocity on the surface of the transducer. mr, Rr, Cm, m, Co, N, and Rm are respectively the radiation mass, radiation resistance, elastic stiffness, mass, clamped capacitance, mechanical–electric conversion coefficient, and fraction force resistance of the transducer’s surface.

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Figure 1. A piezoelectric thin spherical-shell polarized in radial direction.

Figure 2. Two circuits of a spherical-shell transducer: (a) a source and (b) a receiver.

Transmission Response of a Source-Transducer Let’s derive a transmission response function of a source-transducer corresponding to the equivalent circuit, as given in Figure 2(a). As shown in Figure 3, when an external force is applied to a micro-volume of a thin spherical-shell transducer, the radial component of this force can be written as

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Lin Fa and Meishan Zhao

    Fr    2T sin lt rb   2T sin lt rb   . 2 2  

(1)

Assume the material density of the transducer is  , then we have

Fr  lt r 2  

 2ur t 2

(2)

where, u r is a displacement on the surface of the spherical shell. For an infinitesimal volume element,   0 ,   0 , and    ，we have sin   / 2   / 2 and sin   / 2   / 2 , which leads to the equation

of motion in radial direction 

T  T  2 ur  2 rb t

(3)

where T and T are the tangential stresses in  and  directions. The piezoelectric equations of a spherical shell transducer can be written as S  s11ET  s12E T  d31 Er

(4)

S  s21E T  s22E T  d32 Er

(5)

Dr  d31T  d32T   33T Er

(6)

where, S and S are the strain components in  and  directions; Er is the radial component of electric field; Dr is the radial component of electric displacement; sijE , d ij and  ijE are respectively the compliance, piezoelectric and dielectric constants of the piezoelectric material.

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Meanwhile, spherical symmetry provides the simplification conditions T  T , d 31  d 32 , s11E  s22E , and s12E  s21E . Under an external force (F), the planar compliance is given by scE   s11E  s12E  / 2

Newton’s law provides the following equation of motion d  2 ur u F  2r E  31 E Er  2 t  rb sc  rb sc 4 rb2 lt 

(7)

On the surface of a spherical shell S0  4 rb2 , the external force has contributions from two sources F  Fm  Fr

(8)

Figure 3. Micro volume element of spherical shell transducer.

The frictional mechanical-resistance force generated from sphericalshell vibration in a medium with force resistance  is written as

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Lin Fa and Meishan Zhao Fm   S0

dur   Rm vr dt

(9)

The force resistance on surface is Rm  S0 and vr is the vibration velocity. When acoustic waves are radiated outward, the spherical shell transducer in the acoustic field also radiates by itself with an acoustic pressure p . A counterforce is then generated Fr   S0 p |r  r

0

(10)

Assume the transient response of the electric terminals in Figure 2(a) has a zero-state response, then we can define the electric-acoustic conversion function as a ratio of the vibration velocity on surface to the driving voltage signal H1 (s) 

vr (s) ds  3 U1 (s) s  as 2  bs  c

(11)

where a

Rm  Rr 1  m  mr R0C0

b

Rm  Rr 1 N2   (m  mr ) R0C0 (m  mr )Cm (m  mr )C0

c

1 (m  mr )C0Cm R0

d

N (m  mr )C0 R0

From residue theorem, the electric-acoustic impulse response of the spherical shell transducer can be expressed as

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L

h1 (t )   Re  H ( si )e s t 

(12)

i

i 1

where，L is the number of poles of equation (11). The denominator of this function is a cubic polynomial s3  as 2  bs  c  0 which yields the following roots s1  x  y  a / 3

(13)

s2  ( x  y) / 2  a / 3  i 3( x  y) / 2

(14)

s3  ( x  y ) / 2  a / 3  i 3( x  y) / 2

(15)

where

  y   q / 2  D 

x  q / 2  D

1/3

1/3

p  b  a2 / 3

q  c  2a3 / 27  ab / 3

and D  ( p / 3)3  (q / 2)2

Depending on the selected parameter (D), there are three possible cases as the follows: (i). For D  0 ，the cubit polynomial s3  as2  bs  c  0 has three independent

real

roots

(not

complex).

Define A  ( x  y ) / 2 ，

B  ( x  y ) / 2i ， 1  a / 3  2 A , and 1  A  a / 3 , we have

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s1  1 ,

s2   1  3B ,

s3  1  3B

(16)

The electric-acoustic conversion function is then H1 (s) 

ds

(17)

(s  1 )(s  1  3B)(s  1  3B)

The electric-acoustic impulse response function becomes h1 (t )  A1 exp  1t    B1ch 





3Bt  C1sh

This is a case of over damping, where A1  

d1   3B2 2 1

1  1  1 B1 

C1 

d 1

3B  3B 2   12 

d   1  3B 2 

3B  3B 2   12 



exp



exp

sh



3Bt 

ch



3Bt 













3Bt  exp  3Bt 2





3Bt  exp  3Bt 2





3Bt  exp  1t  

(18)

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(ii). For D  0 , the cubit polynomial s3  as2  bs  c  0 has three real roots; two of them are identical roots s1  2 x  a / 3  1

s2、3   x  a  3   1

(19)

The electric-acoustic conversion function is then H1 ( s ) 

ds

 s    s    1

2

.

(20)

1

The electric-acoustic impulse response function is h1 (t )  A2 exp  1t   B2 exp  1t   C2t exp  1t 

.

(21)

This is a case of critical damping, where A2  

B2 

d1

 12

,

d 1  1   1

    1

C2 

2

1

1d 1  1

(iii). For D  0 ， the polynomial s3  as2  bs  c  0 has one real root and two conjugate complex roots s1  2 A  a / 3  1 s2, 3   1  i 3B

(22)

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where A  ( x  y ) / 2 , B  ( x  y ) / 2 , 1  A  a / 3 , and 1  a / 3  2 A . The electric-acoustic conversion function is then

H1 ( s ) 

s   s   1

1

ds



 i 3B s  1  i 3B



(23)

The electric-acoustic impulse response function becomes h1 (t )  A3exp  1t   B3exp  1t  cos 1t  1 

(24)

This is an oscillatory damping solution, where the coefficients A1 and B1 are constants; 1 and 1 are the damping factors; and phase shift, with

1  3B 1  1  1 A3 

d1   3B2 2 1

B3  

C3 

d  1  1 

 12  3B 2

d  11  3B2  3B 12  3B2 

D3  B32  C32

and

1  arctan

 1  3B 2 3B  1  1 

1

is the initial

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As discussed above, even though there are three possible solutions, the oscillatory mode in equation (24) is the only physically meaningful solution for the electric-acoustic transmission function.

Transmission Response of Receiver-Transducer The acoustic-electric conversion of a transducer is an inverse process of electric-acoustic conversion. Following a similar process as discussed above for a spherical shell transducer with harmonic vibration, we have an acoustic-electric equivalent circuit, as shown in Figure 2(b). A physically meaningful solution for acoustic–electric transmission response can be obtained as h3 (t )  A3 exp(-3t )  B3 exp(-3t ) cos(3t  3 )

The coefficients A3 and B3 are constants;

3

(25) and  3 are the damping

factors; and 3 is the initial phase shift. These qualities are determined by physical and geometric parameters of the transducer, as well as the coupling medium around the transducer.

Measurement Set-ups The physical limits of a measurement process must be considered, so to achieve the desired results from a measurement. One of such limits is that in most of the applications we have a reduced order of sinusoidal quantity on time. Another factor which needs special attention is that the values of radiation resistance (R r ) and radiation mass (m r ) in the equivalent-circuits are the functions of signal frequency. Excited by an electric/acoustic signal-wavelet with multi-frequency components, the vibration of the transducer’s surface consists of many sine frequency

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components. Consequently, the impulse response discussed equation (24) is applicable only to the cases of harmonic vibrations. For most cases, both the signal used to excite source-transducer and the signal arriving at the receiver-transducer are wavelets which are composed of multi-frequency components. From Fourier transform, the electric/acoustic-signals can be decomposed into linear combinations of sine-wave components, where each wave component has its own unique amplitude, phase, and frequency. Then, the electric-acoustic/acousticelectric excitation process can be modeled as a parallel-circuit network, as shown in Figure 4. The system in Figure 4 consists of a set of parallel circuits, responsible for electric-acoustic (Figure 4, Parts I) and acoustic-electric (Figure 4, Parts III) conversions. In this system, U1(t) is an electric driving-signal; U3(t) is the measured output electric-signal; h2j(t,ωj) is an acoustic impulse response of the measured object/propagation-medium for jth frequency component; vr3j is the vibration velocity of jth frequency component in the acoustic-signal arriving at the receiver-transducer, originated from vibration velocity vr1j of jth frequency component on the sourcetransducer’s surface; and h1j(t, ωj) and h3(t, ωj) are the electric-acoustic and acoustic-electric impulse responses corresponding to jth frequency component. Each circuit produces its own electric-acoustic and/or acoustic-electric impulse response, owing to its distinctive radiation resistance and radiation mass.

Figure 4. A transmission network in acoustical-measurement process.

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Electric Driving and Received Acoustic-Signals An electric driving-signal wavelet can be expressed by a linear superposition of N frequency components N

N

j 1

j 1

U1 (t )  U1 j (t )   S ( j ) cos  j t   ( j ) 

(26)

The amplitude spectrum S ( j ) and phase spectrum  ( j ) are the components from a N-point discrete Fourier transform, where j = 1, 2, …, N. The measured acoustic-signal, i.e., the output electric-signal at the receiver-transducer’s electric-terminal, is normalized as N

U 3 (t ) 

U j 1

3j

(t ,  j )

 N  max  U 3 j (t ,  j )   j 1 

(27)

where, U 3 j (t ,  j ) is a convolution of the signal functions in the transmission process

U 3 j (t ,  j )  U1 j (t ,  j ) * h1 j (t ,  j ) * h2 j (t,  j ) * h3 j (t,  j )]

(28)

The various properties of a measured object can be obtained by an inversion process from the measured acoustic-signal, i.e., the transducer’s electric-acoustic/acoustic-electric conversion responses. When a driving signal wavelet is used to excite the source-transducer, it produces a response for electric-acoustic transmission. A sample convolution of a gated sine driving electric-signal with the electric-acoustic impulse response from the parallel connected network (Figure 4) is shown in Figure 5. In this example, we used the parameters f1 = 115 kHz and

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t0 = 2/f1. A few selected frequency components of this signal are presented in Figure 5(a-d), i.e., fs = 0.25f1, 0.5f1, 1.0f1 and 5f1. Figure 5(e) is the normalized waveform which is a cumulative convolution of all frequency components in the network. Figure 5(f) is the normalized amplitude of the spectrum.

Figure 5. The convoluted gated-sine driving electric-signal, at f1=115 kHz and t0=2/f1, with electric-acoustic impulse response from the measurement network: (a)-(d) are the selected frequency components at f = 0.25f1, 0.5f1, 1.0f1, and 5f1; (e) is the normalized waveform; (f) is the normalized amplitude spectrum.

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PHYSICAL STRUCTURE The physical structure of this measurement system includes three core modules, namely, a mechanical assembly, an electrical hardware module, and a system software module aimed at control and computation. A flowchart is shown in Figure 6.

Figure 6. Structure flowchart of acoustical measurement system.

Electrical Module A pair of transducers are used as part of the system structure, where, one is a source-transducer and the other is a receiver-transducer. The electric module is an electric-hardware component, composed of this pair of transducers, a micro-controller, an electric-signal waveform generator, a power amplifier, a high resolution/sampling-rate digitizer which is up to 24 bit and 15 MHz, and a computer for central system control.

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Mechanical Modules The mechanical module includes the components of steering-engines, stepping motors, and sliding rails, shown in Figure 7. The combination of this module with a micro-controller forms a positioning platform which is used to slide and/or rotate the source/receiver transducers to suitable position/direction in a silencing tank filled with water. The silencing tank is used to gauge the physical properties of the transducer, e.g., the electric-acoustic/acoustic-electric transient response, directivity, radiation power, receiving sensitivity of the transducer.

Figure 7. Mechanical part of the measurement system for sliding/rotation of a transducer.

Software Module and Interface Display The system software module is developed on LabVIEW platform in Graphic Programming Language (G-code). Communications between a computer and the hardware modules are realized through USB serial ports. Powered with a graphic interface control panel, users have the options in selecting different driving-signals, configuring power amplifier, adjusting the position and rotational angle of the transducers, and attaining data acquisition of the measured acoustic-signal. A typical display penal is shown in Figure 8.

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Figure 8. An interface control penal, showing a waveform of a received acoustic signal.

Figure 9. Flowchart of the software control sub-modules, i.e., electric-signal source, power amplifier, data display/storage, and slippage/rotation.

The developed software module consists of four functional submodules as shown in Figure 9. Under a graphic computer interface, the VISA library from LabVIEW is used to control the four sub-modules through serial port communications. On human-computer interface screen with functional selection, users enter specific parameters to command the modules for special operations and acoustic-measurement. The four software-control functional sub-modules are described as the follows:

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Electric-Signal Source Module In controlling the widgets, knobs, and switches on the software interface panel, this module configures and modulates the types of source signal for the experiment, its frequency and amplitude, and the cycle of an electric source-signal. The options of an electric source-signal include gated-sine wave, square-wave, triangular-wave, and saw-tooth wave. Power Amplification Module This module configures the amplification gain of a power amplifier for an electric-signal generated by an electric-signal waveform generator. The amplified electric-signal is used to excite the source-transducer to radiate acoustic-signal. This is achieved through a rotary button on the interface control panel. For an electric voltage signal with frequency belt from 0.15 MHz to 1.5 MHz, the maximum peak-peak value can reach 220 V. For a voltage signal with frequency range from 10 kHz to 150 kHz, the maximum peakpeak value can be achieved up to 1600 V. For an electric-signal source with general-impedance, an input impedance 50 Ω is generally appropriate. For an electric-signal source with high-impedance, an appropriate input impedance is 50 KΩ. Data Module - Display/Storage/Process From this module, users may configure acquisition channel, sampling rate, and amplitude range of an acquired signal. Figure 8 shows a general display screen for this purpose. Data acquired in time and frequency domains by a digitizer are sent to the interface panel through a USB port. These data can be displayed on screen in real time or stored as data bank for late analysis. The format of the acquired data can be in simple text or in binary form. A MATLAB code has been developed for theoretical analysis of the transient response function of the transducers. The LabVIEW code was developed for the acquisition of the measured data. Comparison of theoretical analysis with the measured experimental data provides a reliable verification of the theory and experimental instrumentation. It

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provides a validation of the new system network-model discussed in this chapter.

Slippage/Rotation Modules This module would automatically initialize the position and direction of source/receiver transducers. The sliding unite moves to default zero position and the steering engines return to the direction of zero degree. Users can configure the stepping distance of the sliding unite and the rotation angle of the steering engine by entering specific parameters on interface control panel. The currently available stepping distance is from 0.0 to 3000.0 mm. The rotating angle of the steering engine is arranged from 0 to 360 degree.

System Workflow Through control panel with graphic interface, users can manage the operation of mechanical and electric hardware, complete data acquisition and on-screen view, and perform numerical calculation, data analysis, and storage of the measured data. The first step is to perform a system alignment through control panel operation. Based on a specific need of a measurement, users may send a specific command to the micro-controller to initialize the system. The stepping motors and steering engines are adjusted on the sliding trails, so to achieve a specific setting in vertical/horizontal direction, as well as a set angle relative to the horizontal plane. This operation will make the source/receiver transducers move to a desired position and rotate to a set direction for specific acoustic-measurement. Now, through control panel, users may send a command to electricsignal waveform generator to produce an electric source-signal, e.g., a gated sine wave, a square wave etc, to the power amplifier. The amplified electric-signal excites the source transducer to emit an acoustic-signal outward. The acoustic-signal then propagates to the receiver-transducer, via a specific medium or through a measured-object.

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Finally, invoking the power of receiving-transducer, the acousticsignal can be converted back to an electric-signal. The final output signal is acquired by a high resolution/sampling-rate digitizer and sent back to the computer for display, analysis, and storage.

AN EXAMPLE Let’s consider an example in practical application. We are going to measure the transient-response of acoustic signals transmitted through spherical thin-shell piezoelectric-transducers. Then, we will compare the experimental results from the measurement to that of the same system from theoretical predictions.

Assumptions and Limits As a test for the measurement system let’s use water as a propagation medium. This is just to simplify the system setting for a basic testing process. Because the acoustic attenuation coefficient of water is very small, we assume with acceptable approximation that an acoustic-signal propagating inside water may have geometrical attenuation but not viscous attenuation. Of course, the geometric attenuation, if too large from a medium, could induce experimental errors if without proper correction. Furthermore, we assume that the acoustic signal propagating inside this medium is elastic. Finally, we assume that all frequency components in the acoustic-signal propagate with the same speed. Obviously, the shapes of waveform and frequency spectra would not change during propagation. The amplitude of the propagating wave would decrease with respect to the propagated distance. Under the conditions mentioned above, the acoustic impulse responses for all frequency range in water are the same that can be written as

Recent Development of an Acoustic Measurement System h2 (t )  h21  t , 1   ...  h2 j  t ,  j   ...  h2 N  t , N  

  t  t1  (1  r )

135

(29)

where, t1 is the propagation time of the radiated acoustic wave from the source-transducer to the receiver-transducer. Let’s consider a gated sine-wave as the source of excitation. A gated sine-waves in time and frequency domains are respectively U1 (t )   H (t )  H (t  t0 )U 0 sin s t 

(30)

and

S1 ( ) 

U 0 s  s cos s t0  j sin s t0  exp   jt0 



2 s

 2 

(31)

where H(t) is a Heaviside unit step function. The parameters are, respectively, the amplitude (U0), the angular frequency (ωs), and the timewindow of the gated sine driving electric-signal (t0).

The System Set-up The measurement system includes two piezoelectric spherical-shell transducers - one for acoustic-source and the other for receiver, a waveform signal generator, a power amplifier, a NI PXI-5922 Digitizer with 16-24 bit and 5-15 MHz sampling rate, a desktop computer, a control central, and a tank with a dimension of 1.5 × 1.5 × 1.3 m3 filled with water as the transmission medium. It is understood that the electric-acoustic conversion of a transducer is reversible which means that it is exactly a reciprocal of acoustic-electric conversion. From a graphic interface control penal with computer automation, one can send a command to the waveform-generator to create a gated sine electric-signal with selected cycles. In the current experiment, we generate

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a signal with two cycles. Going through power amplification, this gated sine electric-signal provides an excitation for the source transducer to emit acoustic-signal. The radiated acoustic-signal propagates to the receiver through the medium and then is converted back to electric-signal through the receiver-transducer. The electric signal can be acquired by a NI PXI5922 Digitizer which provides a feed-back to the human-interface control central. There are different materials which can be used to construct the transducers. For our testing process, the spherical transducers are constructed from piezoelectric material PZT4 and are polarized in radial direction. The physical parameters of piezoelectric PZT4 are given in Table 1. The spherical-shell transducers are constructed with a radius rb = 7.5 mm and a shell thickness lt = 1.5 mm. For the two spherical shell transducers, one is used as source and the other one is used as receiver. With water as the medium around the transducers, a distance between source-transducer and receiver-transducer to set to be 0.7 m. The density of water is taken to be ρm = 1.000 g/cm3 and the propagation velocity for P-wave in water is set as vm = 1428.6 m/s. A two-cycle gated sine-wave electric-signal with fs = 115 kHz was used to excite the source-transducer. Table 1. Physical parameters of PZT4 piezoelectric material T  33  130.4186 1010 F/m2

s11E  12.7  1012 m2/N s12E  4.15 1012 m2/N d31  127  1012 m/V

  7.5 103 kg/m3

Permitivity for dielectric displacement under constant strain, in direction-3 Elastic compliance for stress in direction-1 and accompanying strain in direction-1 Elastic compliance for stress in direction-2 and accompanying strain in direction-1 Induced strain in direction-1 with electric field applied in direction-3 Material density

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Results of Calculations and Measurements The acoustic wave propagated through the water as a medium to reach the receiver-transducer (Figure 4, Part III). Going through acoustic/electric conversion, the receiver transducer converts the transmitted acoustic signal into electric signal. After acoustic/electric conversion, the signals from the receivertransducer are measured and compared to the theoretical predictions. Again, the frequency and time parameters are set at f1 = 115 kHz and t0 = 2/f1. For the purpose of the discussions we selected only a few frequency components to calculate, e.g., f = 0.25f1, 0.5f1, 1. 0f1 , …, respectively.

Figure 10. The convoluted signals at the receiver (Part III of Figure 3) from several selected sine frequency components.

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Figure 10 shows a few selected frequency components of the convoluted-signals calculated at the receiver-transducer’s electricterminals. Both the waveform and the amplitude spectrum of measured acoustic-signal are compared to theoretical predictions in Figure 11. The solid-line in Figure 11(a) is the normalized cumulative outputs from all parallel circuits in Part III of Figure 4, i.e., the normalized waveform of the calculated acoustic-signal from the receiver-transducer’s electric terminals. Figure 11(b) is the corresponding amplitude spectrum. The dotted-lines in Figure 11 are the experimental measurements of the waveform and amplitude spectrum of the acoustic signal measured at the receivertransducer. Figure 11 shows that the experimental results from the measurement network are in generically good agreement with those of the theoretical predictions.

Figure 11. Normalized electric-signals at the receiver-transducer (Part III of Figure 4) with fs = 115 kHz and t0=2/f1 converted by a spherical shin-shell transducer. The solid lines are from the theoretical model and the solid dotted-lines are from experimental measurement: (a) waveform; (b) amplitude spectrum.

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CONCLUSION The first question we might ask is what we have achieved from this research. Well, there are several aspects we might want to summarize here. Overall, the most important, we have developed a new acoustic measurement system and it is practically feasible and accurate. Established on the platform of an acoustic-signal transmission network, we have developed a new acoustical-measurement system, which is based on the parallel-connected circuit-network model for acousticmeasurement. The influences on the measurement from system hardware and the environment are considered and rectified. These factors include but are not limited to the propagation medium, the physical and geometrical parameters of the transducers which are responsible for the quality of electric-acoustic conversion and vice versa, and specific characteristics of a driving electric-signal. With a proper parameterization and rectification of each of the influencing factors, the system can be practically accurate. The system can be used either for new experiments or be used to test the reported measurements from other research, e.g., inversion analysis of a studied object from the data of theoretical calculation, inversion analysis of experimental measurement, determination of acoustic parameters of a measured object, determination of the properties of transducers, e.g., directivity, radiation power, receiving sensitivity, etc. Practical operations of this acoustic-measurement system reveal advanced features, e.g., high measurement accuracy following-on high sampling rate and resolution, easy to operate with user friendly graphic human-computer interface, simple hardware structure owing to instrument virtualization and high level of integration, and easiness for functional expansion. Now, what have we learned from this work? We have used this newly developed circuit-network measurement system for a prototype application and used the measured data to test theoretical predictions of the transientresponses from the piezoelectric transducers. A detailed examination of the measured experimental data reveals that there are quite spaces for improvement.

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Figure 11 shows that the measured acoustic-signal waveform from the receiver-transducer is very much like that of theoretical calculation, but not quite the same. The central frequency of the calculated acoustic-signal at the receiver electric-terminals is at 116.67 kHz, whereas central frequency of the measured acoustic-signal is 115.77 kHz. Comparing to theoretical calucations, the normalized amplitude spectrum of the measured acousticsignal at the receiver-transducer’s electric-terminals is shifted slightly to the low frequency domain. The percentage error between theoretical calculation and experiment measurement is about 0.77%. We believe that this discrepancy comes largely from the assumption of the propagation medium. Realistically, water has a certain degree of viscosity which may lead to attenuation of the acoustic-signal during the propagation process. To achieve a desired accuracy, there are other potential sources of errors, other than trasmission media, which may need additional attention. In most of the cases of practical applications, the driving electric-signal/acousticsignal contains many frequency components. The radiation impedance of a transducer is quite largely influenced by the signal frequency. The properties of electric-acoustic conversion of a transducer is not only determined by its physical/geometrical parameters but also the property of the driving signal. Therefore, in order to model a transient response properly, the frequency factor must be properly managed. Going through the process of modeling, theoretical calculation, and experimental measurement, we have obtained an enhanced understanding for the transient responses of the electric-acoustic/acoustic-electric conversions of piezoelectric transducers. In summary, the transient response of a spherical thin-shell transducer can be described by a harmonic sine-model based on the principle of linear supposition. The cumulative signal at the receiver, contributed from multiple frequency components, provides a transient reality. Therefore, the acoustic measurement process can be modeled realistically as a signal transmission network composed of many parallel-connected equivalent-circuits. The question is how to use what we have learned from this work for our continued research to achieve higher accuracy for acoustic measurements,

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and to develop even higher level of accurate acoustic measurement instrumentations. There are numbers of open issues associated with the development of highly desirable acoustic measurement systems. One fundamental issue is to eliminate effectively the potential errors induced from the various physical parameters associated with the system hardware in the experiment set-up. Another issue is to extend the current measurement system for applications beyond the current setting. In a recent review, Zhao and Fa [31-39] discussed different aspects of the acoustic measurement in various practical industrial applications. It remains challenging to apply the discussed measurement system for all these industrial applications. Understandably, the network model that we have discussed in this chapter would need significant extension.

ACKNOWLEDGMENTS This work is supported in part by Xi’an University of Posts and Telecommunications, National Natural Science Foundation of China (Grant no. 41974130), and the Physical Sciences Division at The University of Chicago.

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[33] Zhao, Y., Zhao, N., Fa, L., and Zhao, M. (2013). “Seismic Signal and Data Analysis of Rock Media with Vertical Anisotropy,” Journal of Modern Physics 4: 11-18. [34] Fa, L., Tian, Y., Xie, W., and Zhao, M. (2012). “Effects of ElectricAcoustic and Acoustic-Electric Conversions of Transducers on Acoustic-Logging Signal”, Chin. Sci. Bulle. 57: 1246-1260. [35] Fa, L., Zhao, M., Castagna, J. P., Liu, Y. C., Wang, L., Wang, Y. Q., and Sun, J. G. (2014). “On Polarization of Plane Wave Propagating inside Elastic Hexagonal System Solids,” Science China: Physics, Mechanics & Astronomy 57:1-12. [36] Fa, L., Wang, L., Liu, L., Zheng, Y., Zhao, Y., Zhao, N., Li G., and Zhao, M. (2013). “Research Progress in Acoustical Application to Petroleum Logging and Seismic Exploration,” The Open Acoustics Journal 6: 1-10. [37] Fa, L., Qi, Z., Wenhui, C., Ding, P. F., Liang, M., Mou, J. P., Tang, J. J., Zhang, Y., Zhao, M. (2019). “Theoretical study and experimental Verification of a transient response model of thin spherical shell piezoelectric-transducer,” Journal of Vibration Engineering xx: xx. [38] Fa, L., Fa, Y., Zhang, Y. D., Ding, P. F., Gong, J., Li, G. H., Li, L., Tang, S. J., and Zhao, M. (2015). “Anomalous incident-angle and elliptical-polarization rotation of an elastically refracted P-wave,” Scientific Reports 5: 12700. [39] Fa, L., and Zhao, M. “Recent progress in acoustical theory and applications.” in Understanding Plane Waves. xxx: Nova Science, 2019 (companion chapter in this book).

In: Understanding Plane Waves Editor: William A. Cooper

ISBN: 978-1-53616-779-5 © 2020 Nova Science Publishers, Inc.

Chapter 3

PLANE NONLINEAR ELASTIC WAVES: APPROXIMATE APPROACHES TO ANALYSIS OF EVOLUTION Jeremiah Rushchitsky* Department of Rheology, S. P. Timoshenko Institute of Mechanics, Kyiv, Ukraine

ABSTRACT Three approaches (methods) are used to analyze the evolution of the plane longitudinal and transverse waves that are propagated in a nonlinear hyperelastic medium - method of successive approximations, method of slowly varying amplitudes, method of restriction on the displacement gradient. The evolution is understood in the standard for physics meaning: the wave with some given initial profile (harmonic or solitary) evolves, that is, changes this profile. The term “the profile is distorted” is used sometimes too. The medium of propagation is described by the wellknown in nonlinear mechanics of materials five-constant Murnaghan model. First, the noted methods are described briefly as applied to the *

Corresponding Author’s Email: [email protected].

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Jeremiah Rushchitsky wave propagation problems. Further, the longitudinal and transverse plane waves are analyzed separately. The point is that the Murnaghan model describes the longitudinal wave by the quadratic nonlinear wave equation, whereas the transverse wave – by the cubic nonlinear wave equation. Ten variants (known and new = published and unpublished) for the longitudinal wave and four variants for the transverse wave (known and new = published and unpublished) of an approximate analysis are described and commented on. It is shown that each variant provides an answer on a certain aspect in the initial wave profile evolution study. A statement of some variants is accompanied by the 2D and 3D pictures. An attention is drawn to the features of the evolution process as well as to similarities and differences in the results obtained.

Keywords: longitudinal hyperelastic wave, Murnaghan’s nonlinear model, distortion of initial wave profile, variants of approximate analysis of wave evolution

INTRODUCTION The plane waves (including elastic ones) are most fully studied [1-8]. An analysis of such waves in the framework of the linear approach can be considered complete and is expounded in the university text-books on the linear theory of elasticity [9-13]. However, the nonlinear analysis is still in a state of development and is in progress in different directions [8, 14-16]. A fragment of research in one of these directions is described below. At that, the analysis of plane waves will be presented. Among the classical types of plane elastic waves, the plane longitudinally polarized and transverse vertically polarized waves (P-wave and SV-wave) are chosen and an evolution of the initial profile of such waves is studied. It is well-known that the model of linear elastic deformation does not describe an evolution of the waves (waves do not interact) [1-13]. So, the nonlinear models should be used. Three nonlinear models are utilized most frequently – Neo-Hookean, Rivlin-Mooney, Murnaghan [8, 17-20]. They describe both the geometrically nonlinear strains, and the physically

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nonlinear ones. First two are applied to the materials that are undergone the large (finite) deformations. The Murnaghan’s model is one of the most developed models of nonlinear elasticity theory, it describes well the deformation of many engineering materials in the wide range of strains – from small to large. Therefore, this model is used in the next considerations for the analysis of evolution. Within this framework, the propagation of plane waves under the condition that they move in the direction of the abscissa axis is described by the nonlinear wave equations of the different order of approximations that include the orders from 2 to 5 [8]. The case of saving the nonlinearity of the 2nd order only is characterized by the following quadratic nonlinear wave equations for three polarized plane elastic P-, SH-, SV-waves  u1,tt     2  u1,11  N1 u1,11u1,1  N 2  u2,11u2,1  u3,11u3,1  ,

(1)

 u2,tt   u2,11  N 2  u2,11u1,1  u1,11u2,1  ,

(2)

 u3,tt   u3,11  N 2  u3,11u1,1  u1,11u3,1  ,

(3)

N1  3    2    2  A  3B  C   , N 2    2   1 2  A  B ,

(4)

 is the density, uk is the displacement;  ,  , A, B, C are the elastic constants of the Murnagan’s model. The good feature of nonlinear wave equations (1)-(3) is that the lefthand sides are the classical linear wave equations whereas the right-hand sides include the quadratically nonlinear summands only. This structure of equations turned out very convenient in future studies. The second novelty consists in that in contrast to the linear wave equations the nonlinear ones are the coupling equations and this coupling isn’t symmetric. This means that the P-wave can be self-generated and be in this way distorted owing presence in (1) of the nonlinear term N1 u1,11u1,1 . It can be also generated by the SH- and SV-waves owing presence of the nonlinear terms

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N 2  u2,11u2,1  u3,11u3,1  . As it can be seen from (2), (3), the SH- and SV-waves

cannot be self-generated and the evolution of wave initial profile cannot be studied in this approach. Also, the transverse waves cannot be generated by the longitudinal one. This is possible when initially P-wave and SH or SV-waves) are excited simultaneously. In the case when only one of these waves is initially excited, the following equation should be solved  u1,tt     2  u1,11  N1 u1,11u1,1 ,

(5)

u2,tt  u2,11  0 , u3,tt  u3,11  0 .

(6)

As can be seen from equations (5), (6), the longitudinal wave is described by the nonlinear wave equation and the transverse waves are described by the linear wave equation. Thus, the evolution of P-wave can be studied within the quadratically nonlinear approximation, whereas the study of evolution of the SH- (SV-) wave needs the next (cubic) approximation. The case of saving the nonlinearity up to the 3rd order is characterized by the following quadratically and cubically nonlinear wave equations for three polarized plane elastic P, SH , SV  waves [8]  u1,tt     2  u1,11  N1 u1,11u1,1  N 2  u2,11u2,1  u3,11u3,1    N3 u1,11  u1,1   N 4  u2,11u2,1u1,1  u3,11u3,1u1,1  , 2

u2,tt  u2,11  N 2  u2,11u1,1  u1,11u2,1    N 4 u2,11  u2,1   N 5 u2,11  u1,1   N 6 u2,11  u3,1  , 2

2

2

u3,tt  u3,11  N 2  u3,11u1,1  u1,11u3,1    N 4 u3,11  u3,1   N5 u3,11  u1,1   N 6 u3,11  u2,1  , 2

2

2

(7)

(8)

(9)

Plane Nonlinear Elastic Waves N3 

151

3    2    6  A  3B  C  , 2

N 4  1 2   2    2    5 A  14 B  4C 

(10)

N5   3 2   2  A  2B  , N6  3 A  10B  4C.

The cubic system (7)-(9) permits to mark out at least seven new possibilities (as comparing with the quadratic system (1)-(3)) in analysis of the nonlinear wave interaction [8]. But one only possibility is important for the following study of evolution in this chapter. The presence of summands

N

4

u2,11 (u2,1 ) 2  ,  N 4 u3,11 (u3,1 ) 2 

in equations of the transverse wave

propagation (8), (9) testifies that in the used approach the harmonic transverse wave will generate its own third harmonic. It is worthy to note once again that in the quadratic representation (2),(3) (in which the summands mentioned above are absent) the initially given transverse wave in the form of the 1st harmonic generates the same harmonic and therefore does not describe the wave evolution. Note also that equations (8),(9) are identical with exactness to indexes (2 must be changed on 3 and vice versa) what is true for equations (2),(3) too. This means that any transverse wave can be chosen separately. Let the SV-wave will be chosen in the following. Thus, the P-wave will be studied in following on the base of the wave equation (5) with quadratically nonlinear right side

 u1,tt     2  u1,11  N1 u1,11u1,1  u1,tt   cL  u1,11   N1   u1,11u1,1 , 2

where cL 

   2  

(11)

is the P-wave velocity in the linear model.

The SV-wave will be studied in following on the base of the wave equation with the cubically nonlinear right side

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 u3,tt   u3,11  N 4 u3,11  u3,1  . 2

(12)

The following part of this chapter includes a comparative analysis of many (existing and new) variants that are based on equations (11) or (12) and described an evolution of the initial wave profile. It should be noted at once that each variant responds to a certain aspect in the wave evolution study of P-wave or SV-wave.

2. ABOUT THE METHOD OF SUCCESSIVE APPROXIMATIONS This method is called sometimes the perturbation method or the method of small parameter [7. 8]. The introduction of small parameter  is the main feature of this method. Let us narrow the description of method to the theory of elasticity and consider the displacements vector u  x, t  . Assume that it is sufficiently smooth and apply this method to equation (5). According to the method, the function u  x, t ,   is sought in the form of convergent series 

u ( x, t ,  )   nu ( n ) ( x, t )  u (0) ( x, t )   u (1) ( x, t )   2u (2 ( x, t ) 

(13)

n0

The zeroth term u (0)  x, t  is assumed to be the solution of the corresponding to (5) linear wave equation

u1,tt   cL  u1,11  0 . 2

(14)

Note that an existence of the linear part of nonlinear equation is a necessary condition of this method. The solution of nonlinear equation (5) is sought in the form of sequential approximations

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

u ( x, t )  u ( x1 , t ,   1)   u ( n) ( x1 , t )  u (0) ( x, t )  u (1) ( x1, t )  u (2 ( x1, t ) 

.

(15)

n 0

A feature and characteristic advantage of this method is that the arbitrary appro ximation u ( n )  x1 , t  is found as the solution of inhomogeneous linear equation ( n) ( n1) ( n1) u1,( ntt)   cL  u1,11   N1   u1,11 u1,1 , 2

(16)

that is, to find the n -th approximation, it is necessary to know only the

 n  1 -th

approximation and solve only the inhomogeneous linear wave

equation. This method works well in the theory of waves, when the initial amplitudes don’t increase essentially (in some cases, half as much again).

3. ABOUT THE METHOD OF SLOWLY VARYING AMPLITUDES This approximate method was proposed by Baltazar van der Pol for problems of nonlinear oscillations in radio-physics [2, 8, 21, 22]. Sometimes it is called the van der Pol method. Then it was widely used in mechanics. The basic hypothesis is that the solution of the weak nonlinear wave equation is close to the linear solution. The term “slowly varying amplitude” means that the wave amplitude should vary very slowly over a distance equal to one wavelength. The base for analysis is formed by the linear solution in the form of harmonic wave. Therefore, the nonlinear solution is looking for the form u1 (x, t)  A1 (x )e 

i k1 x t 

or

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



u1 ( x, t )  Re A1 ( x)ei k1x t   a1 ( x)cos k1 x  t  1 ( x) .

(17)

The procedure of solving is such that further the interaction of some finite number (mainly, two, three, four) of waves (17) is studied. Then the solution of the basic nonlinear equation is assumed as M

u1 ( x1 , t )   A1m ( x1 )ei m , m 1

 m  k1m x1  mt.

(18)

Let us narrow analysis to the nonlinear wave equation (5). Then the next few steps should be as follows: 1. Substitute of (18) into (5). 2. Take into account that (18) is the solution of a linear equation. 3. Neglect the second derivatives because of the energy external flux is assumed to be absent. 4. Save only products of amplitudes for taking into account of an interaction and self-interaction of waves-participants. As a result, the shorten (reduced, abridged) equations will be obtained i  k1m  A1m ,1 e m   M

m 1

M M i (  ) N1 k1n k12p A1n A1 p e n p .  2(  2 ) n 1 p 1

(19)

Further, some additional assumptions relative to relationships between wave numbers k1m and frequencies m should be done. Assume now that three waves only are studied. Then the first assumption is called the condition of frequency synchronism (matching) 1  2  3 .

(20)

In this case, the shorten equation splits up into three evolution equations i k13  k12  k11 x1

 A11 ,1   1 A12 A13e

,

Plane Nonlinear Elastic Waves i k13  k12  k11  x1

 A12 ,1   2 A11 A13e

i k13  k12  k11  x

 A13 ,1   3 A11 A12 e   

155

,

(21)

,

N1 k1 k13 (k1  k13 ) Nk k ;     1 11 12 ; 2(  2 )k1 2(  2 )

(    3).

Usually, the evolution equations (21) are analyzed, when the second matching condition - wave numbers matching - is assumed k11  k12  k13.

(22)

Studying the evolution equations is a separate problem accompanying the use of the slowly varying amplitude method. This method works successfully when the basic nonlinear wave effects - self-influence (selfgeneration), energy pumping from one wave to other ones, self-switching and so on – must be described.

4. ABOUT THE METHOD OF RESTRICTION ON DISPLACEMENT GRADIENT The first feature of this method [23-25] is that the nonlinear wave equation to which it is applied should have the special structure. Just equations (5) and (12) have such a structure: the right-side part of the equation can be carried over to the left hand-side one and then formally the nonlinear equation can be written as the linear wave equation with the variable wave velocity. In the case of equation (5) (that is, in the case of quadratic approximation for the P-wave), the appropriate form of equation has the form

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



2 u1,tt   vL    N1   u1,1 u1,11  0  u1,tt  v u1,11  0, 2

(23)

v  vL 1   u1,1 ,    N1    2  . Further, the initial profile of wave is assumed to be the arbitrary function that can describe both harmonic, and solitary waves

u  x1, t  0  F  ax1  ,

(24)

where a is the arbitrary parameter characterizing the wave length for harmonic waves and wave bottom for solitary waves. Then the standard wave phase variable   a  x1  vt  is introduced and it is assumed that the wave is propagated in the form

u  x1 , t   F   .

(25)

The root in a representation of the variable velocity v  vL 1   u1,1 is further written in the form of series with very important restriction (this can be meant as the second feature of this method)

 u1,1

1.

(26)

This restriction permits to represent approximately the velocity by only two first approximations

1   u1,1  1  1 2 u1,1 .

(27)

Then the phase can be approximately written as follows

  a  x1  vLt   1 2 avLu1,1t 

(28)

And the solution can be represented also approximately in the form

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157

u1  x1 , t   F  a  x1  vLt  1 2  vLu1,1t   

(29)

 F  a  x1  vLt   1 2  avLu1,1t 

Now, the solution (29) can be expanded into the Taylor series by the small parameter

  1 2 avLu1,1t

(30)

1

within the neighborhood of the classical constant value   a  x1  vL t  and with saving only two first terms u1,1  x1 , t   F       x1  aF   , u1  x1 , t   F   F,1/   a  F    1 2 a 2vLt  F / ,1   . 2

(31)

The approximate representation (31) of solution (29) has a general character. It describes for different initial profiles F  x1  the one and the same nonlinear effect – the initially linear form of solution (the 1st harmonic in the case of harmonic profile or the initial solitary wave profile) is complemented by the generated by nonlinearity of material part of solution (the 2nd harmonics in the case of harmonic profile). From the point of view of wave mechanics, a presence of this new part means a distortion of the wave initial profile, because the value of this part is increased with time of wave propagation. In this way, the plane wave is evolved. Further 16 variants of studying the evolution of the initial wave profile will be shown.

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5. VARIANT 1 (P-WAVE, CLASSICAL HARMONIC PROFILE, METHOD OF SUCCESSIVE APPROXIMATIONS, FIRST TWO APPROXIMATIONS) The initial harmonic wave profile is defined by the formula

u1  x1 , t  0  u1o cos kL x1

(32)

( u1o is the initial wave amplitude, k L is the wave number). It is considered that the corresponding wave has the form u1  x1 , t   u1o cos  k L x1  t 

(33)

(  is the wave frequency). In the analysis of equation (5) by the method of successive (1) approximations, the 1st approximation u1  x1 , t  is assumed to be the

solution of the corresponding linear wave equation (14)

u1,tt   cL  u1,11  0 2

and coincides with The 2 linear

nd

(34). (2) 1

approximation u

wave

equation

 x1 , t 

is a solution of the inhomogeneous

(2) (1) (1) u1,(2)tt   vL  u1,11   N1   u1,11 u1,1

2

(34)

and

is

nd

characterized by the 2 harmonic [8] 

 2 N1 u1o  kL2  cos2  kL x1  t .   8   2  

u1(2)  x1, t   x1 

(35)

Plane Nonlinear Elastic Waves

159

Thus, the approximate solution corresponding to variant 1 consists of the sum of two harmonics [8] u1V 1  x1 , t   u1(1)  x1 , t   u1(2)  x1, t  

  2 N1  u1o cos  kL x1  t   x1  u1o  kL2  cos2  kL x1  t .   8   2  

(36)

The solution (3) confirms theoretically a generation of the 2nd harmonic. According to (36), this nonlinear wave effect is formed in three stages. Initially, the wave profile is slightly different from the linear harmonic wave profile. Further, with increasing the distance, which passed the wave, or the time of wave propagation, the 1st harmonic is summed up with the 2nd harmonic, whose amplitude is slowly increasing, and they together form a modulated wave. Step by step, an influence of the 2nd harmonic increases and it becomes the dominant one. Thus, variant 1 describes an evolution of the wave as a gradual transition from a profile in the form of the 1st harmonic to the profile of the 2nd harmonic. The short characteristic dependence of the amplitude of P-wave on the time and distance of its propagation is shown in Figure 1 (the SI-system is used). Note. All plots in this chapter are built for metallic materials, small srains, and ultrasound frequencies. These plots are made in different mathematical packages (Mathematika, MathLab, etc.). The 2D picture shows an evolution in coordinates “distance of propagation – displacement” and the 3D picture – in coordinates “time of propagation – distance of propagation – displacement.” Because this variant is probably the most studied before, then below the stages of evolution are shown more in detail in 2D pictures [8].

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Figure 1. Evolution of the initial harmonic profile: in coordinates “distance of propagation – displacement” (2D picture) and– in coordinates “time of propagation – distance of propagation – displacement” (3D picture).

The profile evolution is divided into four stages: Stage 1. The initial harmonic profile it tilts downwards under the constant angle, i.e., the maximal positive values decrease and the maximal negative values increase. That is, on this initial stage the profile is transformed from the harmonic one into not harmonic, but very close to harmonic profile. Stage 2. The tops of the profile get lower, and the plateau is gradually formed instead of the peak. Later the plateau lowers even more, the middle part of plateau begins to sag and the profile becomes two-humped instead of one-humped. The frequency of repetition of the same profile is equal to the initial oscillation frequency. Stage 3. Saving the prior frequency the profile becomes more clearly two-humped with an increasing sag up to the point when it touches abscissa axis.

Plane Nonlinear Elastic Waves

161

Stage 4. The sag increases and the profile becomes similar to harmonic one with the 2nd harmonic frequency but with the unequal amplitude swings: upwards – the large amplitude, downwards - roughly half the size of the prior one, upwards - slightly bigger than the prior upper one, downwards - roughly twice as big as the prior lower one. So, the gradual change (progress) of the 1st harmonics profile transforms it into the 2nd harmonic profile and we can observe the transformation of 1st harmonics into the 2nd one. Figures 2 – 5 shows consequently the mentioned above stages. 0.04

22

0

24

26

28

Figure 2. The faintly developed stage of evolution. 0.05 0

17

18

19

20

21

22

23

Figure 3. The poorly developed stage of evolution. 0.1

17

0

18

19

20

21

22

18

19

23

Figure 4. The intermediately developed stage of evolution. 0.5

0

14

15

Figure 5. The critical stage of evolution.

16

17

20

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Let's consider below the variants 2 and 3 and compare the first three variants on the ground that they may be very close in describing of evolution the harmonic wave.

6. VARIANT 2 (P-WAVE, CLASSICAL HARMONIC PROFILE, METHOD OF RESTRICTION ON THE GRADIENT OF DISPLACEMENT, FIRST TWO APPROXIMATIONS) Method of restriction on the gradient of displacement is applied to the nonlinear equation (5), which is transformed into the form of a linear equation with the variable velocity of wave propagation (11). After some transformations and restrictions, an approximate solution can be written in the form of (31).

Figure 6. Evolution of the initial harmonic profile: in coordinates “distance of propagation – displacement” (2D picture) and– in coordinates “time of propagation – distance of propagation – displacement” (3D picture).

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163

Let the initial wave profile is harmonic in space and described by the function F  x1   eikL x1 . Then formula (31) takes a more concrete form of the modulated wave u1  x1 , t   a o e

 ikL  x1 cLt 

 1 2  cLt  k L   a o  e 2

2

2ik L  x1 cLt 

.

(37)

The characteristic dependence of the amplitude of the wave (37) on the time and distance of the wave propagation is shown in Figure 6. The 2D picture shows an evolution in coordinates “distance of propagation – displacement” and the 3D picture – in coordinates “time of propagation – distance of propagation – displacement.”

7. VARIANT 3 (P-WAVE, CLASSICAL HARMONIC PROFILE, METHOD OF SLOWLY VARIABLE AMPLITUDES) The starting equations are three evolution equations (21). The stated problem on the 2nd harmonic generation is the simplest one. It is assumed here that the 3rd wave is initially not excited, that is, A3  0   0 . It is also assumed that the 1st and 2nd waves are identical (all parameters are constant and known, equal in the amplitude, wave number and frequency). Then it is necessary to solve only the 3rd equation of system (21). The corresponding solution has the form

N k  2 A3  x1    1 L1  A1  x1 2    2  2

(38)

So, with the simplifications adopted, the 3rd wave has at outlet of medium the frequency 21 and wave number 2kL1 , that is, it is the 2nd harmonic for the 1st wave. At that, the amplitude of the 3rd wave increases in direct proportion to the distance x1 that passed the wave. It also depends

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on the squared amplitude and the squared wave number of the 1st wave, what means that this amplitude can in some cases increase very quickly. This problem is often used as an example in the nonlinear optics as one that adequately describes experiments on the 2nd harmonic generation. The classical experiment with the optical wave is as follows: the red light of a ruby laser turns into the purple light when dihydrogen phosphate ammonia is passed through the crystal [21]. Thus, this variant describes such a fact: after a certain time from the beginning of the wave motion in the form of the 1st harmonic, the 2nd harmonic joins the 1st one with the amplitude that arises due to the mechanism of self-generation of the wave and increases with time propagation of the wave. Note finally, that an appearance in the final solution of the 2nd harmonic is common to variants 1-3. This harmonic is gradually superimposed on the 1st one and after some time becomes dominant. The difference between these variants can be explained by the different velocity of evolution due to various restrictions on the smallness of the wave parameters.

8. VARIANT 4 (P-WAVE, BELL-SHAPED PROFILE, METHOD OF SUCCESSIVE APPROXIMATIONS, FIRST TWO APPROXIMATIONS) A bell-shaped profile has the form of the Gauss’s function [8, 25], and therefore the wave with such a profile has, according to the general representation (24), the form u1  x1 , t   u1o e 

2

2

.

(39)

By the method, the 1st approximation is a solution of the corresponding to (5) linear equation and has the form (24). The 2nd approximation can be searched as the solution of an inhomogeneous equation

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165

(2) (1) (1) u1,(2)tt   vL  u1,11   N1   u1,11 u1,1 2

(40)

(2)   N1    u1o  a3 1   2  e . or u1,(2)tt   vL  u1,11 2

2

2

(41)

An attempt to find a solution of equation (41) in the form u1(2)  A   e  shows that this form is the solution of a corresponding 2

homogeneous equation (d’Alembert’s wave). Therefore, the solution should be complicated in this way u1(2)  t A   e  . 2

(42)

By substituting the representation (42) into the left part of equation (41) and the change B    t A   e  , the differential equation relative to 2

function В   can be obtained  B    3  2  1 B      u1o 2  a 1   2  e  

2

/2

.

(43)

The homogeneous equation

B    3  2  1 B    0

(44)

corresponds to equation (11) from subsection (2.173) of [26] with a 0, b 3, c  3 and can be reduced to the so-called Whittaker equation. At that, a choice of the corresponding to the right hand side of (43) partial solution seems very complicate owing presence the factor

  2  1 . Thus, the 2nd approximation will have a complicated mathematical form, which should still be found and, accordingly, analytical and numerical analysis of wave evolution within the framework of the first two approximations is looking very unpromising for the profile

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in the form of the bell-shaped function. In this situation, some advantage has the method of restriction on the gradient of displacement.

9. VARIANT 5 (P-WAVE, BELL-SHAPED PROFILE, METHOD OF RESTRICTION ON THE GRADIENT OF DISPLACEMENT, FIRST TWO APPROXIMATIONS) This method is described in version 2, the profile - in version 4. It is sufficient to write the final formula (31) for the first two approximations [24, 25] u1  x1 , t   Ao e

2

2

 1 2  t cL a 2 2  Ao  e  . 2

2

(45)

First of all, the bell-shaped wave evolves and the profile is distorted symmetrically due to the appearance of a "2nd harmonic" whose amplitude increases nonlinearly with the propagation time of the wave. Note. The term “harmonic” should be used for bell-shaped waves sufficiently conditionally. If to assume that the next harmonics after the 1 st harmonic e

2

2

are distinguished only by the coefficient (simple number)

before the phase and this number gives the number for harmonic, then e can be called the 2nd harmonic. But the notion “harmonic” is used nevertheless in the harmonic analysis that is based on the completeness of the functions-harmonics. This completeness is not valid for the bell-shaped functions. Therefore, here the term “2nd harmonic” has to be written with inverted commas. Note. The mathematical simplicity of obtaining the 2nd harmonic is due to the possibility to use only the 1st derivative of the Gauss function. The characteristic dependence of the wave amplitude on the time and distance of wave propagation is shown in Figure 7. On the upper picture, the initial profile is superimposed on the distorted one. The bottom picture is three-dimensional, which shows evolution both in time and in space. 2

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Figure 7. Evolution of the initial symmetrical bell-shaped profile: in coordinates “distance of propagation – displacement” (2D picture) and– in coordinates “time of propagation – distance of propagation – displacement” (3D picture).

A comparison with the change in the harmonic wave profile shows that the bell-shaped wave changes its profile in a slightly different way. The “2nd harmonic” always gives off a negative additive. So, the slopes of the distorted “bell” become steeper. The upper part of the “bell” falls and form two “bells.” Since the wave is elastic and energy losses cannot be, then the change in the profile is consistent with the law of conservation of wave energy when the wave propagates.

10. VARIANT 6. (PROFILE IN THE FORM OF THE WHITTAKER FUNCTION, THE METHOD OF RESTRICTION ON THE GRADIENT OF DISPLACEMENT, FIRST TWO APPROXIMATIONS) The Whittaker function Wk ,m ( z ) is defined for all values k , m and for all real nonnegative z [26]. It can be presented on through a degenerate hypergeometric function  ( a, c, z )

168

Jeremiah Rushchitsky 

z

Wk ,m ( z )  e 2 z

m

1 2

1     m  k  ,2m  1, z  . 2  

(46)

The method is described in variant 2, the initial wave profile is chosen as follows

F  x   aoW1/4;1/4  x  .

(47)

Then formula (31) has more concrete form

u  x, t   a0W1/4;1/4    1 2  t cL a 2  a0  W /1/4;1/4    2

 a0W1/4;1/4    1 2  t cL a  a0  2

2

2

2

 1 1   .  4  2  aW1/4,1/4     

(48)

The form of solution (48) testifies that it describes a changing the initial profile of the solitary wave due to the presence of a nonlinear component and the direct dependence of this component on time and, more specifically, describes the initial profile "spreading". The resulting formula (48) can be used for numerical simulation of wave evolution. The characteristic dependence of the wave amplitude on the time and distance of wave propagation is shown in Figure 8. In the upper picture, the initial profile (lower curve) is superimposed on the distorted. The bottom picture is three-dimensional, which shows evolution both in time and in space. Thus, the solitary wave with an asymmetric profile in the form of Whittaker’s function evolves in three directions: the initial amplitude decreases, the left and right parts of the hump gradually become steeper, the profile itself gradually becomes more symmetrical, resembling a bellshaped profile. The wave’s bottom in all cases remain unchanged.

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Figure 8. Evolution of the initial asymmetric profile: in coordinates “distance of propagation – displacement” (2D picture) and in coordinates “time of propagation – distance of propagation – displacement” (3D picture).

11. VARIANT 7 (P-WAVE, CLASSIC HARMONIC PROFILE, SEQUENTIAL APPROXIMATION METHOD, FIRST THREE APPROXIMATIONS) The profile and method are described in variant 1. It is here convenient to present a solution in the framework of the first two approximations in the form [8]

u1(12)  x, t   u1o cos  u1o Mx1 cos2 ,

M

N1 k2 1 1 2 2 u1o  kL   N1u1o L2  N1u1o 4 . 8  2  8 vL 8 vL

(49)

(50)

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Jeremiah Rushchitsky

According to the method, the 3rd approximation is the solution of the next equation (3) (2) (2) u1,(3)tt   vL  u1,11   N1   u1,11 u1,1 . 2

(51)

This solution is as follows [8]    4  5  11 3 3  8  u1(3)  u1o M L   x1      cos 4  .  sin 4     2 2 3 2 k x 3 8  k L   x1    L 1    

(52)

Thus, the 3rd approximation introduces the 4th harmonic into the solution. Thereafter, the 4th approximation will introduce the 8th harmonic (at each step, the harmonics are doubled). The solution within the first three approximations has the form [8]\ u2(1 23)  x1 , t   u1(0) x1 , t   u1(1) x1 , t   u1(2) x1 , t   u1o cos   u1o M L x1 cos 2   8   4  5 11 3 3 u1o  M L   x1     sin 4      cos 4  . 2 2  3 8 k   x    3 2k L x1  L 1  

(53)

The main wave effect is that first the wave shape differs slightly from the 1st harmonic, although the 2nd and 4th harmonics already affect the profile shape. With increasing the distance that the wave passed, the influence of the 2nd harmonic increases and it becomes dominant, but further the 4th harmonic becomes the dominant one. This fact changes the understanding of the 2nd harmonic as the main one, which corresponds to the theoretical results of variants 1-3 and experiments on optical waves. It can be assumed that an experiment on a longer distance will show, in certain cases, the dominance of the 4th harmonic. The following plots (Figures 9-18) show ten stages in the progress of evolution and demonstrate well the briefly described before evolution. On all the figures, the lower plot corresponds to the first two approximations, the upper plot corresponds to the first three approximations in two variants – with allowance for two last summands in

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171

formula (53) or without these summands. Both variants coincide on all the plots, what testifies the assumed before proposal on the small effect of these summands on the general wave picture. These plots show the graceful lowering of the plot to the negative values of amplitude, what has been fixed before in the analysis within the framework of the first two approximations. So, with time the change of amplitude occurs periodically (the motion is wave-like) relative to some negative value of amplitude. 1

0.5

5

10

15

20

25

30

120

125

130

220

225

230

-0.5

-1

Figure 9. Amplitude versus distance (stage 1). 1

0.5

105

110

115

-0.5

-1

Figure 10. Amplitude versus distance (stage 2).

0.5

205

210

215

-0.5

-1

Figure 11. Amplitude versus distance (stage 3).

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Jeremiah Rushchitsky

0.5

245

250

255

260

265

270

290

295

300

-0.5

-1

Figure 12. Amplitude versus distance (stage 4).

0.5

275

280

285

-0.5

-1

Figure 13. Amplitude versus distance (stage 5). 0.5

305

310

315

320

325

330

520

525

530

-0.5

-1

Figure 14. Amplitude versus distance (stage 6). 1 0.5

505

510

515

-0.5 -1 -1.5

Figure 15. Amplitude versus distance (stage 7).

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173

1.5 1 0.5

605

610

615

620

625

630

770

775

780

820

825

830

-0.5 -1 -1.5

Figure 16. Amplitude versus distance (stage 8). 2

1

755

760

765

-1

Figure 17. Amplitude versus distance (stage 9).

2

1

805

810

815

-1

Figure 18. Amplitude versus distance (stage 10).

As it was mentioned above basing on the general considerations, the 1 harmonic dominates up to about the amplitude value M * z  1 4  , which st

corresponds to reaching by the 2nd harmonic of the amplitude value equal to 1/4 of the 1st harmonic amplitude. But the effect of plot lowering is displaying at once and is presented on all the plots. The allowance for the 4th harmonic accelerates the process of lowering, what is displaying starting about with M * z  1 5 . Within the framework of the 2nd and 3rd approximations in the range M * z  1 4;1/ 3 the plot forms initially the plateau at the upper pick, further

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Jeremiah Rushchitsky

the cave is forming at the plateau and the tendency to forming the 2 nd harmonic is displaying. In the case of restriction to the 2nd approximation, this tendency is aggravating and the 2nd harmonic is still dominating. But allowance for the 3rd approximation changes the tendency and after reaching the amplitude value M * z  1 3 the 4th harmonic becomes dominant. The Figures 15 and 16 show the formation of two caves instead one and one hump in the middle of caves. These changes demonstrate as if the 4th harmonic formation, because the lower pick is yet not deformed. When the value M * z   3 4 being only reached (Figure 18), then first the plateau and further the hump are forming at the lower pick. This testifies that the 4th harmonic is forming.

12. VARIANT 8 (P-WAVE, ARBITRARY WAVE PROFILE F  x1  , METHOD OF RESTRICTION ON THE GRADIENT OF DEFORMATION, FIRST THREE APPROXIMATIONS) The method is described in variant 2. Let us remind some facts from this variant. It is assumed that the wave is propagated in the form

u  x1, t   F   , where v  vL 1   u1,1 is the variable velocity of the nonlinear wave. Next steps will be new. The root is expanded into a series for a small parameter with saving of not the first two members, but three ones

1   u1,1  1  1 2  u1,1  1 8 2  u1,1  . 2

(54)

The approximate representation (54) can be considered as preserving the first three approximations in the analysis of nonlinear wave equation (23). Since the wave velocity is now written as 2 v  vL 1  1 2  u1,1  1 8 2  u1,1   , (55) an approximate solution can be  

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175

represented in a form that generalizes the corresponding representation from variant 2





u1  x1 , t   F a  x1  vLt   1 2  t vLu1,1 1  1 4  u1,1  .

(56)

The assumption about the smallness of the quantity

    1 2  t vLu1,1 1  1 4  u1,1 

(57)

enables a decomposition of function (56) into a Taylor series for a small parameter   in the neighborhood of a phase value   a  x1  vL t  u  x1, t   F       F    F /     1 2 F //   2 

(58)

And save only the first two members





u1  x1 , t   F   F,1/   a 2 1 2  t vL F,1/   1  1 4  aF,1/      F    1 2  a 2vLt  F / ,1   1  1 4  aF,1/    . 2

(59)

Thus, the solution within the framework of three approximations (59) includes an additional factor versus (7), which introduces antisymmetric changes in the form of initial profile to the characteristic of a profile of (7) symmetric form of distortion. The amplitude of antisymmetric distortion can be determined by numerical calculations, which needs the values of physical constants of engineering materials that can be found in several scientific publications (in particular, in [19-21]). It should be noted that representation (59) contains the 3rd approximation of cubic nonlinearity, which in the case of a harmonic profile means the presence of the 3rd harmonic in contrast to the effect of variant 7, where after the 2nd harmonic the 4th one is generated.

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13. VARIANT 9 (P-WAVE, HARMONIC WAVE PROFILE, METHOD OF RESTRICTION ON THE GRADIENT OF DEFORMATION, FIRST THREE APPROXIMATIONS) This variant can be meant as a continuation of variant 8 for the concrete harmonic profile. The starting formula is (59) with u  x1 , t   F    u1o cos  u1o cos  k L x1  t  ,

(60)

F,1/    u1o kL sin  .

(61)

Then solution has the form of three summands u1  x1 , t   u1o cos  1 2  t  u1o k L a  vL sin 2  1  1 4  au1o k L sin   . 2

(62)

Thus, the solution (62) includes the first three harmonics – 1st, 2nd, and 3rd. The characteristic scenario of evolution in four stages is shown in Figure 19. The upper line corresponds to the initial harmonic profile, the middle to the first two approximations, the lower line – to first three approximations. Thus, the middle line shows the tendency of transition to the 2nd harmonic, whereas the upper line shows the influence of the 3rd approximation. As a result, the 3rd approximation changes essentially the picture of evolution – instead characteristic for the first two approximations two humps, three humps appear quite clearly. At that, the maximal amplitude of resulting profile increases as compared with the case of the first two approximations. It should be noted that here the difference in the description of evolution by the method of successive approximations and the restrictions on displacement gradient is displayed once again – the 3rd approximation

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introduces into the solution the 4th harmonic in the method 1 and the 3rd harmonic in the method 2. u 1. 10

6

5. 10

7

u

0.01

5. 10

7

1. 10

6

0.02

0.03

0.04

1. 10

6

5. 10

7

x

0.51

5. 10

7

1. 10

6

Stage 1 6

1. 10

6

5. 10

7

5. 10

7

5. 10

7

1. 10

6

1.5 10

6

1.01

1. 10

6

0.54

x

u

1. 10

7

0.53

Stage 2

u

5. 10

0.52

1.02

1.03

1.04

x

Stage 3

2.01

2.02

2.03

2.04

x

Stage 4

Figure 19. Evolution of the initial harmonic profile in four stages.

14. VARIANT 10 (P-WAVE, PROFILE IN THE FORM OF THE GAUSS FUNCTION, METHOD OF RESTRICTION ON THE GRADIENT OF DISPLACEMENTS, THE FIRST THREE APPROXIMATIONS) The profile is described in version 6, the method - in variant 2. Substitution of the expression



F,1/    Aoe

2

2



 a Aoe

2

2

(63)

,1

into the general formula (59) gives a solution in the form of three terms

178

Jeremiah Rushchitsky u1  x1 , t   Aoe  Ao e

2

2

2

2

 1 2  t cL a 2 2  Ao  e 1  1 4  avL Ao e  2

2

 1 2  t cL a 2 2  Ao  e   1 8  t 2 a 3  vL   Ao   3e 3 2

2

2

3

2

2

.

2

2

 

(64)

If conditionally the expression e 2 is assumed to be the first harmonic, then solution (64) includes the first three harmonics. An evolution is now described in the more complicate form and accelerated as compared with version 5. The characteristic scenario of evolution in three stages is shown in Figure 20. Stage 1 shows two lines. The internal line corresponds to the initial bell-shaped profile and the external one corresponds to the first two approximations. This stage shows a tendency in formation two humps from one hump (two “bells” instead of one “bell”). This is looking very similar to the case of the harmonic profile, when the 1st harmonic tends to the 2nd one. 2

A

A

0.005 0.006

0.004 0.003

0.004

0.002 0.002

0.001

9.97

9.98

9.99

10.00

m

24.97

Stage 1

24.98

24.99

Stage 2

A 0.010 0.008 0.006 0.004 0.002

39.97

39.98

39.99

40.00

Stage 3. Figure 20. Evolution of the initial Gaussian profile in three stages.

m

25.00

m

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179

The stage 2 and 3 corresponds to the 2nd and 3rd approximations. The red line corresponds to the 2nd approximation and shows some transformation of the symmetric profile into the asymmetric one. Whereas the 3rd approximation works in opposite way – it tries to return the profile into the symmetric state. Thus, the 3rd approximation changes very slightly the profile evolution that is formed by the first two approximations.

15. VARIANT 11. (P-WAVE, PROFILE IN THE FORM OF WHITTAKER'S FUNCTION, METHOD OF RESTRICTION ON THE GRADIENT OF DISPLACEMENTS, FIRST THREE APPROXIMATIONS) The profile is described in version 6, the method - in version 2. The formalism of obtaining the solution is the same as in variant 9. The substitution of the expression  1 1 F,1/     a0W1/4;1/4   ,1  a0 a   W1/4,1/4   (65) into formula (36)  4 2 

gives a solution in the form of three terms u  x, t   a0W1/4;1/4  x   1 2  t cL  a   a0  W /1/4;1/4  x    2

2

2

2

1  2  1   a0W1/4;1/4    t cL a 4  a0    1W1/4,1/4    3 2     3

1  3  1   t 5cL  a0    1W1/4,1/4   . 16  2  

(66)

If conditionally the expression W1/ 4,1/ 4   is the “1st harmonic”, then the solution (66) includes the first three “harmonics”. An evolution of the

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Jeremiah Rushchitsky

wave profile is now described in the more complicate form and accelerated as compared with the approach from version 6. 2.2 10

6.6 10

7

7

0.000035 0.00008

0.00003 0.000025

0.00006

0.00002 0.00004

0.000015 0.00001 5. 10

0.00002

6

x 0.0011 0.0012 0.0013 0.0014 0.0015 0.0016

x 0.0031 0.0032 0.0033 0.0034 0.0035 0.0036

Figure 21. Evolution of the initial Whittaker’s profile.

The characteristic scenario of evolution in two pictures is shown in Figure 21. The left picture 1 shows two lines. The internal line corresponds to the initial bell-shaped profile with the maximal value of amplitude 3.3 10-6 and the external one cor responds to the first two approximations. This stage shows a tendency to the significant increase the non-symmetric hump to the more symmetric shape. The right picture 2 shows the profile of the 1st +2nd approximations (external line) and the profile of the 1st + 2nd +3rd approximations (internal line). Thus, the 3rd approximation slightly returns the maximal amplitude to the initial shape saving at that the tendency to more symmetrical shape.

16. VARIANT 12 (SV-WAVE, CLASSICAL HARMONIC PROFILE, METHOD OF SUCCESSIVE APPROXIMATIONS, FIRST TWO APPROXIMATIONS) The basic nonlinear wave equation is equation (12)

 u3,tt   u3,11  N 4 u3,11  u3,1  , 2

(67)

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for which the corresponding linear wave equation has the form  u3,tt   u3,11  0 .

(68)

Further, according to the method of successive approximations, the small parameter has to be introduced into the slightly changed equation (67)

u3,tt   vT  u3,11   N4   u3,11 u3,1  . 2

2

(69)

The next steps in this approach are described in variant 2. But here some difference will be in the form of the recurrent equation (n) ( n 1) u3,( ntt)   vT  u3,11   N 4   u3,11 u3,1(n1)  . 2

2

(70)

Owing to (70), the solution within the framework of two first approximations is represented as follows u3  x1 , t   u3(0)  x1 , t   u3(1)  x1 , t  .

(71)

The first summand in (71) represents the linear soluton, which for the harmonic profile with the given initial amplitude u3o and frequency  has the form u3(0) ( x, t )  u3oe  T

i k xt 

  k   v  . T

(72)

T

The solution (72) describes the linear wave effect: the wave in the form of 1st harmonic propagates without distortion and does not interact with itself. To obtain the next approximation, it is necessary to solve the equation (1) u3(1)  vT2u3,11  i  N4   u3o  kT3e 3

3i kT x1 t 

.

(73)

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The right-hand side of (73) includes the 3rd harmonic, what testifies the case of parametric resonance. Therefore, the solution of the inhomogeneous equation will be as follows u3(1) ( x1, t )   N4 6 u1o  kT2 x1e 3

3i kT x1 t 

.

(74)

The solution in the form of the first two approximations has the form

u3  x, t   u1o ei k x t   x1  N1 6   u1o  kT2  e3i k x t  .   T 1

3

T 1

(75)

This solution can be commented in the way that it confirms theoretically the generation of the 3rd harmonic. This nonlinear wave effect is formed in three stages. First, the profile differs slightly from the initial one. Further, with increasing the distance of wave propagation or time its propagation, the 1st harmonic is summed with the 3rd harmonic, amplitude of which increases gradually, and they form the modulated wave. Step by step, an effect of the 3rd harmonic increases and it becomes the dominant one. The situation is very similar to the case of P-wave with harmonic profile within two first approximations.

17. VARIANT 13 (SV-WAVE, CLASSICAL HARMONIC PROFILE, METHOD OF SLOWLY VARIABLE AMPLIITUDES, FIRST TWO APPROXIMATIONS) This method is described in subsection 3. The starting nonlinear wave equation is equation (69). The nonlinear solution is represented in the form

u3 ( x1 , t )  A3 ( x1 )ei kT x1 t 

(76)

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Consider now an interaction of three waves. Then the solution can be written as a sum of waves with the variable amplitudes u3 ( x1 , t )  A31 ( x1 )ei1  A32 ( x1 )ei 2  A33 ( x1 )ei 3 ,  m  kTm x1  mt.

(77)

When the standard for this approach procedures are performed, then the three coupled ordinary differential equations of the first order relative to amplitudes are obtained

3

k  A  m 1

Tm

3 m ,1

e

i m

k  k 2 k A A A ei n  2 p  q     Tn Tp Tq 1n 1 p 1q .    N 4      2 i n  p  2 q  n 1 p 1 q 1   k k   Tn Tp  kTq  A1n A1 p A1q e  3

3

3

(78)

Further, the conditions of frequency synchronism must be introduced

1  22  3  0, 21  2  3  0 .

(79)

Then the shorten equations (78) are transformed into three coupled evolution equations

 A31 ,1  1  A32 

2

2

i k13  k12  2k11 x1 A33e ,

2

i kT 3  kT 2  2kT 1 x1 A32e ,

 A32 ,1   2  A31   A33 ,1   3  A31 

i kT 3  2kT 2  kT 1 x1 A33e ,

(80)

N k  k N k k (k  k )     4 T T 3 T T 3 ;  3   4 T 1 T 2 ; (    3).  kT  2

Now, the additional assumption on the synchronization of wave numbers should be formulated kT 3  kT 2  2kT1  0, kT 3  2kT 2  2kT 1  0 .

(81)

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The conditions (81) are needed for an analysis of different problems on an inter action of waves-participants. The simplest problem is the problem on the generation of the 3rd harmonic. It is assumed here that the 3rd wave is initially not excited, that is, A33  0   0 . It is assumed also that the 1st and 2nd waves are identical (by amplitudes, wavenumbers, and frequencies). Finally, it is assumed for coordination with the classical case on the generation of the 2nd harmonic that the amplitude of the 1st wave is initially constant A31  const . As a result, the 3rd equation of the system (80) only must be solved A33,1   3  A31  . 3

(82)

The corresponding solution is as follows

A33  x1     N4   kT 1   A31  x1 . 3

3

(83)

Thus, with the simplification described above the 3rd wave has at outlet from a medium of propagation the frequency 31 and wave number 3kT 1 . That is, it is the 3rd harmonic for the 3rd wave and its amplitude increases directly proportional to the distance x1 passed by the wave. This amplitude depends also on the cubed amplitude A31 and cubed wave number kT 1 of the initial wave. To obtain the necessary parameters of the generated wave at outlet, these three quantities can be controlled. At that, it is necessary to remember that the quantity N4  characterizes the elastic medium (material) and its change has the narrow frame. So, such a problem is described theoretically: the harmonic wave is generated at the entrance into the medium with cubically nonlinear mechanical properties; this wave is self-generated while being passed through the medium; at the outlet from the medium, this wave is transformed into the 3rd harmonic of the initial wave.

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18. VARIANT 14 (SV-WAVE, CLASSICAL HARMONIC PROFILE, METHOD OF RESTRICTION ON DISPLACEMENT GRADIENT, FIRST TWO APPROXIMATIONS) The starting nonlinear wave equation is (67) or (69). Then equation (69) is transformed into the following one







u3,tt   vT    N 4    u3,1  u3,11  0  u3,tt  1  3  u3,1  2

2

2

 v  u 2

T

3,11

 0,

(84)

where 3   N4   . Assume further that the initial profile od the wave is described by the formula u3  x1 , t  0  F3  ax1  ,

(85)

where by the arbitrary parameter a the wavelength or wave bottom can be controlled. In the next step, the wave is assumed to be propagated in the form u3  x1 , t   F3   ,   a  x1  vt  ,

(86)

where the wave velocity is the variable one v  1  3  u3,1  vT . 2

(87)

Further, it seems to be rational to suppose the restriction in (87)  3  u3,1 

2

1,

what allows to write the root in the form of series

(88)

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

1  3  u3,1   1  3  u3,1  2



2 1/2

 1  1 2 3  u3,1   1 832  u3,1   2

4

.

(89)

Then the condition (88) permits to represent approximately the variable velocity (87) by the first two summands and to write the solution as follows





2 u3  x1 , t   F3 a x1  vT t  1 2 3  u3,1  t  .  

(90)

Note. An exactness of approximation (90) depends on exactness of fulfilling the condition, which includes the restriction on two parameters: parameter and squared 3  (  2 ) /    (5 A  14B  4C) / 2  displacement gradient  u3,1  . 2

On the value of parameter  3 . The mentioned above restriction can be linked with the class of engineering materials, for which the Murnaghan’s model is quite acceptable. For metals (aluminum, copper, iron) the following values can be found

3  58 ( Al );  46 (Cu); 113 (Fe) . On the small elastic strains. The classical restriction for small strains us as follows u1,1

1

(91)

(that is, the displacement gradient is assumed to be small). Now, the wave phase   a  x1  vt   a  x1  1   3  u3,1  vT t  must be   2

considered and represented approximately as follows 2    a  x1  vT t   1 2  3avT  u3,1  t  .  

(92)

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187

The formula (92) shows that the phase consists of two parts: the classical phase with the constant phase velocity  clas  a  x1  vT t  and possibly small additional parameter

   1 23avL  u3,1  t . 2

(93)

Because parameter (93) is really small for the sufficiently small time and small wave bottom, then solution (90) can be expanded into Taylor series by the small parameter  in the neighborhood of the value of classical wave phase  3 u3  x1 , t   F3  clas     F3  3   F3/  3   1 2  F3//  3  2 

. (94)

When the condition of the smallness of parameter  is adopted    1 2  3avT  u3,1  t 2

1,

(95)

then in (94) the two first summands only can be saved u3  x1 , t   F3  3     F3  3   F3/  3  .

(96)

Note. Because the smallness of   u3,1  is already assumed in (88), 2

then (95) is the condition on the passed by wave distance avT t . Let us calculate approximately the derivative through the function F3 u3,1  x1 , t   F3,/       x/1 





 F3,/  3  a  F//  3   1 2  a 2vT  u3,11  t  F3,/  3 

Then the formula (96) can be represented in the form

.

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u3  x1 , t   F3  3   F3/  3   F3  3   1 2  3vT t  F3/  3  . 3

(97)

The approximate representation of solution (97) has the general character and for different chosen initial profiles describes one and the same nonlinear wave effect – generation in addition to the 1st harmonic the 3rd harmonic (or similar summands in the case of not harmonic waves) and increase of amplitude of this new summand with time of wave propagation.

19. VARIANT 15 (SV-WAVE, SOLITARY GAUSSIAN PROFILE, METHOD OF SUCCESSIVE APPROXIMATIONS, FIRST TWO APPROXIMATIONS) The initial wave profile is chosen as the Gaussian like (39) u3  x1 ,0   u3o e



 x12 2



(98)

and write the solitary nonperiodic wave in the form u3  x1 , t   u3o e



  32 2



,

(99)

The wave (99) represents the concrete form of the simple D’Alembert wave [31-33] and fulfills the linear equation (68) for the SV-wave. According to the method of successive approximations, the 1st approximation is as follows u3(1)  x1 , t   u3o e



  32 2



.

(100)

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The 2nd approximation u3(2)  x, t  is found as the solution of inhomogeneous equation (2) (1) u3,(2)tt   c3  u3,11   N 4   u3,11 u3,1(1)   2

2

(101)

(2) u3,(2)tt   c3  u3,11   N4    u3o 1   3  e 

2

2



  32 2

 

o  u3 3e 



  32 2



2

  

3 2 (2) 2 2 3  2 u3,(2)tt   vT  u3,11    N4   u3o   3  1   3  e   . 2 3

(102)

3 2  Since the presented at the right-hand side (102) “3rd harmonic” e is the solution of the homogeneous equation (68), then the solution should be sought as the solution of resonance type 2

3

u3  x1 , t   u3(1)  x1 , t   u3(2)  x1 , t   u3oe



 2 2



 x1  N 4   u3o 

3

 2 1   2 

31    3

2



e



3  2 2



This situation was commented in variant 4. Let us repeat this comment here. A choice of the corresponding to the right hand side of (102) partial solution seems very complicate owing presence the factor   2  1 . Thus, the 2nd approximation will have a complicated mathematical form, which should still be found and, accordingly, analytical and numerical analysis of wave evolution within the framework of the first two approximations is looking very unpromising for the profile in the form of the bell-shaped function. In this situation, some advantage has the method of restriction on the gradient of displacement.

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20. VARIANT 16 (SV-WAVE, SOLITARY GAUSSIAN PROFILE, METHOD OF RESTRICTION ON DISPLACEMENT GRADIENT, FIRST TWO APPROXIMATIONS) The profile is the same as in prior variant, the method is decribed many times before, the general form of solution is given by formula (97). Then the solution of the stated in the recent variant will have the concrete view u3  x1 , t   u3oe

   x v t 

 o

2

1

T

2

2

 1 2  t3vT  o   x1  vT t   u3o  e 3

3

3

   x v t 

3  o

2

1

T

2

2

. (103)

This solution depends essentially on the classical wave phase (phase with the constant linear phase velocity) vT )    o  x1  vT t  : in different points of the profile, this profile is changed differently, but the nonlinear summand is always asymmetric relative to bell top. At the point   0 (bell top) the change is absent, that is, the maximal amplitu de of the profile is invariable. In other symmetric relative to the bell top points, the profile is changed non-symmetrically: the right-hand side of the profile is extended (the profile becomes fuller) and the left-hand side is narrowed (the profile becomes thinner). The characteristic dependence of the amplitude of the wave (103) on the time and distance of the wave propagation is shown in Figure 22. The 2D picture shows an evolution in coordinates “distance of propagation displacement” and the 3D picture - in coordinates “time of propagation distance of propagation - displacement”. Thus, the plots above show the initial stages of evolution only, when the plateau starts to form and the maximal amplitude rests unchanged. At that, the left-hand side of the bell is raised and the right-hand side of the bell begins to fall. The maximal amplitude and bottom of the bell are unchanged.

Plane Nonlinear Elastic Waves

Stage 1

Stage

191

Stage 2

3 Stage 4

Figure 22. Evolution of the initial asymmetric profile: in coordinates “distance of propagation – displacement” (2D picture, four stages of evolution) and– in coordinates “time of propagation – distance of propagation – displacement” (3D picture).

So, the shown in pictures evolution is essential and change of initial wave para-meters are essential, too. A comparison with the analogical case for the P-wave testifies some distinctions in the evolution of P- and SVwaves. First of all, the basic nonlinear wave equations are different, because they include nonlinearity of the 2nd order for the P-wave and the 3rd order for the SV-wave. This is manifested in different effects. For example, the speed of evolution for the SV-wave is usually less of the speed for the P-wave. Also, the P-wave shows the symmetrical relative to the bell top evolution, whereas the SV-wave shows the nonsymmetrical evolution. The scenarios of evolution are therefore very different – for the Pwave, the bell shows a tendency to the transformation into two symmetric

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bells, and for SV-wave, the bell shows a tendency to the transformation into two asymmetric bells.

GENERAL CONCLUSION The nonlinearity of plane elastic waves is studied within the Murnaghan’s five-constant model. This generates specific nonlinear wave equations, which are different for the longitudinal and transverse plane waves. Therefore, the evolution of these waves is developed according to different scenarios. All three applied methods - sequential approximations, slowly varying amplitudes, restrictions on the gradient of displacement - are well adapted to the analysis of wave evolution with a harmonic profile. The distortions of more complex profiles in the form of Gauss and Whittaker functions are analyzed successfully by the method of restriction on the gradient of displacement. The wave representation within the framework of two and three first approximations and the corresponding evolution of the wave are distinguished significantly when being studied by the methods of successive approximations and restriction on the displacement gradient. The method of successive approximations describes the generation of the 1st, 2nd, 4th, 8th, etc. harmonics, while the method of restriction on the gradient of displacement describes all the harmonics consecutively – the 1st, 2nd, 3rd, etc. This generates different scenarios of wave evolution. The study of an evolution of plane waves is the multi-parameter problem. It inclu des three groups of parameters – maximal wave amplitude, length or bottom of a wave, six mechanical properties (density and five elastic constants) of material. These parameters cannot be arbitrary. For example, an assumption on smallness of strains restricts the ratio of maximal wave amplitude to the length or bottom of wave – it must be sufficiently small. Also, the length or bottom of a wave and the distance of wave propagation must be rational (must have the physical sense).

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REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]

Achenbach, J. 1973. Wave Propagation in Elastic Solids. Amsterdam: North- Holland. Bedford, A. and Drumheller, D. 1994. Introduction to Elastic Wave Propagation. Chichester: John Wiley. Harris, J. 2001. Linear Elastic Waves. Cambridge Texts in Applied Mathematics. Cambridge: Cambridge University Press. Lempriere, B. 2002. Ultrasound and Elastic Waves: Frequently Asked Questions. New York: Academic Press. Rabinovich, M. and Trubetskov, D. 1989. Oscillations and Waves in Linear Systems. New York: Kluwer Academic Publishers. Royer, D. and Dieulesaint, E. 2000. Elastic Waves in Solids (I,II). Advanced Texts in Physics. Berlin: Springer. Rushchitsky, J. 2012. Theory of Waves in Materials. Copenhagen: Ventus Publishing ApS. Rushchitsky, J. 2014. Nonlinear Elastic Waves in Materials. Series: Foundations of Engineering Mechanics. Heidelberg: Springer. Atkin, R. and Fox, N. 1980. An Introduction to the Theory of Elasticity. London: Longman. Hahn, H. 1985. Elastizitätstheorie (Theory of Elasticity). Stuttgart: B. G. Teubner. Lur’e, A. 1999. Theory of Elasticity. Series: Foundations of Engineering Mechanics. Heidelberg: Springer. Nowacki, W. 1970. Teoria sprężystośći (Theory of Elasticity). Warszawa: PWN. Timoshenko, S. and Goodier, J. 1970. Theory of Elasticity. 3rd ed. New York: McGraw Hill. Maugin, G. 1999. Nonlinear Waves in Crystals. Oxford: Oxford University Press. Engelbrecht, J. 2015. Questions about Elastic Waves. Berlin: Springer.

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[16] Nassar, H., Chen, H., Norris, A., Haberman, M. and Huang, G. 2017. “Non- reciprocal wave propagation in modulated elastic materials” Proceedings of the Royal Society A, 473(2202): 20170188. [17] Guz, A. 1999. Fundamentals of the Three-dimensional Theory of Stability of Deformable Bodies. Series: Foundations of Engineering Mechanics. Berlin: Springer. [18] Holzapfel, G. 2006. Nonlinear Solid Mechanics. A Continuum Approach for Engineering. Chichester: Wiley. [19] Murnaghan, F. 1967. Finite Deformation in an Elastic Solid. 2nd ed. New York: John Wiley. [20] Ogden, R. 1997. The Nonlinear Elastic Deformations. New York: Dover. [21] Yariv, A. 1967. Quantum Electronics. New York: John Wiley. [22] Scott, A. 1970. Active and Nonlinear Wave Propagation in Electronics. New York: John Wiley. [23] Rushchitsky, J. and Yurchuk, V. 2016. “One Approximate Method for Analyzing Solitary Waves in Nonlinearly Elastic Materials” International Applied Mechanics 52(3): 282-290. [24] Yurchuk, V. and Rushchitsky, J. 2017. “Numerical Analysis of Evolution of the Plane Longitudinal Nonlinear Elastic Waves with Different Initial Profiles” International Applied Mechanics 53(1): 104-110. [25] Cattani, C. and Rushchitsky, J. 2007. Wavelet and Wave Analysis as applied to Materials with Micro and Nanostructure. Singapore: World Scientific. [26] Gradstein, I. and Ryzhik, I. 2007. Table of Integrals, Series, and Products. 7th revised edition, edited by Jeffrey, A. and Zwillinger, D. New York: Academic Press Inc. [27] Olde Daalhuis, A. 2010. “Confluent Hypergeometric Functions. Whittaker Functions. Chapter 13.” In Handbook of Mathematical Functions, edited by Olver, F. W. J., Lozie, D. W., Bousvert, R. F. and Clark, C. W., NIST (National Institute of Standards and Technology), Cambridge: Cambridge University Press.

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[28] Kamke, E. 1977. Differentialgleichungen. Lösungmethoden und Lösungen. (Differential Equations. Methods of Solving and Solutions) Wiesbaden: Vieweg+Teubner Verlag, Springer Fachmedien Wiesbaden GmbH. [29] Rushchitsky, J., Cattani, C. and Sinchilo, S. 2005. “Physical constants for one type of nonlinearly elastic fibrous micro- and nanocomposites with hard and soft nonlinearities” International Applied Mechanics 41(12):1368–1377. [30] Lur’e, A. 1990. Nonlinear Theory of Elasticity. Amsterdam: NorthHolland. [31] Hauk, V. (editor) 1997 (e-variant 2006). Structural and Residual Stress Analysis. Amsterdam: Elsevier Science B. V.

BIOGRAPHICAL SKETCH

Prof. D.Sc. Jeremiah J. Rushchitsky Head of the Department of Rheology S. P. Timoshenko Institute of Mechanics of the National Academy of Sciences of Ukraine, Kyiv, Ukraine Personal Data - Born in Ukraine; Nationality: Ukrainian; Citizenship: Ukraine; Languages: Ukrainian, Russian, Polish, English, German.

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Education - Secondary school, 1948-1958, graduated with silver medal; Lviv State University, 1958-1963, graduated with distinction in 1963; Institute of Mechanics, Kyiv, postgraduate study, 1965-1968. Academic Degrees - Dipl.-Eng. (Diploma with Distinction), 1963; Candidate of (Physical-Mathematical) Sciences (PhD), 1968, conferred byTaras Shevchenko, Kyiv University; Doctor of (Physical-Mathematical) Sciences (DSc), 1982, conferred by Mikhail Lomonosov, Moscow University; Professor, 1987, Physical-Mathematical Faculty, National Technical University of Ukraine, “Kyiv Polytechnic Institute”; Member of the National Academy of Sciences of Ukraine, 2018. Employment - 1963-1965, service with the colours Soviet Army, soldier; 1965-present, postgraduate student, junior scientist, senior scientist, head of laboratory, head of department, Institute of Mechanics; 1987-present, professor, Chair of Mathematical Analysis, Physical-Mathematical Faculty, National Technical University of Ukraine, Kyiv; 1994, visiting professor, University of Erlangen-Nürnberg, Germany; 2002- 2006, 2008 visiting professor, University of Rome “La Sapienza”, Italy; 2003, 2004, 2007- 2010 visiting professor, University of Aberdeen, Scotland, UK; 2004- 2006, 2008 visiting professor, University of Salerno, Italy. Publications - Author of 560 publications, including: 12 monographs on waves and composite materials and 5 university text-books on waves in materials, viscoelasticity, wavelet analysis, fractal analysis; about 260 journal papers, 6 State Standards and so on. Recent books: Cattani C., Rushchitsky J. J. Wavelet and wave analysis as applied to materials with micro- or nanostructure. Singapore-London: World Scientific Publishing Co. Pte.Ltd., 2007. – 466 p. (sold about 450 books, cited 136 times); Guz I. A., Rushchitsky J. J., Guz A.N. Mechanical Models in Nanomaterials. In: Handbook of Nanophysics. In 7 vols. Ed. K. D. Sattler. Vol. 1. Principles and Methods. Boca Raton: Taylor & Francis Publisher (CRC Press), 2011. – 827 p.; Rushchitsky J. J. Theory of waves

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in materials. Copenhagen: Ventus Publishing ApS, 2011– 270 p. (free textbook in BookBooN.com, to present downloaded 700,000 times); Guz A. N., Rushchitsky J. J. Short introduction to mechanics of nanocomposites. – Rosemead, CA: Scientific & Academic Publishing, 2013. – 280 p.; Rushchitsky J. J. Nonlinear Elastic Waves in Materials. Heidelberg: Springer, 2014.– 454p. (to present chap-ters of this book downloaded 13,000 times; at present is translated into Chinese and published 2018 by Beijing Institute of Technology Press). Last chapter in book: Rushchitsky J. Cylindrical Surface Wave: Revisiting the Classical Biot’s Problem, Chapter in the book “Seismic Waves – Probing Earth System” Eds. Ono D. and Monoto H. Singapore-London: InTechOpen, 2019. 300 р. P.201-220 Last key journal publications: 1. Developing the mechanical models for nanomaterials. Composites. Part A: Applied Science & Manufacturing. 2007, co-authors I. A. Guz, A. A. Rodger, A. N. Guz. 2. Comparative computer modeling of carbon polymer composites with carbon or graphite microfibers or carbon nanotubes. Computer Modeling in Engineering & Sciences. 2008, co-authors I. A. Guz, A. N. Guz. 3. Revisiting displacement functions in three-dimensional elasticity of inhomogeneous media. Int. J. Solids and Structures, 2009, co-author Kashtalyan M. 4. Fragments of the theory of nanotransistors: self-switching of the plane transverse hypersound wave in composite materials. Int. Appl. Mech., 2009. 5. On constructing the foundations of mechanics of nanocomposites. Int. Appl. Mech., 2011, co-author Guz A.N. 6. Modeling nanocomposites of сomplex shape. Int. Appl. Mech., 2011, co-author Guz A. N. 7. Certain class of nonlinear hyperelastic waves: classical and novel models, wave equations, wave effects. Int J. Appl. Math. & Mech., 2013. 8. Some fundamental aspects of mechanics of nanocomposite materials, J. of Nanotechnologies, special issue “Nano-composites 2013”, 2013, co-author Guz A.N. 9. On features of continuum description of nanocomposite material. J. of Research in Nanotechnology, 2014, co-author Guz A.N. 10. Auxetic linearly elastic isotropic materials: restrictions on elastic moduli,

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Archive of Appl. Mech., 2015. 11. Auxetic metamaterials from position of mechanics: linear and nonlinear models. Reports of Nat. Acad. of Sci. of Ukraine, 2018, N7. In scientific journals of the most prestigious academies of sciences – Comptes Rendus de l'Academie des Sciences, Serie Mecanique; Philosophical Transactions of the Royal Society A: Мathematical, Physical and Engineering Sciences; Atti dell'Accademia Peloritana dei Pericolanti, Classe di Scienze Fisiche, Matematiche e Naturali - French, English and Italian, respectively. In leading world scientific journal of the highest rating –Applied Mechanics Reviews, Composites. Part A: Applied Science and Manufacturing, Composites Sciences and Technology, International Journal of Solids and Structures, Zeitschrift für Angewandte Mathematik und Mechanik - ZAMM, International Applied Mechanics, Computer Modeling in Engineering and Sciences, International Journal of Nonlinear Sciences and Physical Simulation, Mechanics of Composite Materials, Mathematical and Computer Modelling in Dynamical Systems, International Journal of Applied Mathematics Mechanics, Archive of Applied Mechanics. Hirsch-index 25 (1394 citations), Egghe-index 41(1685 citations), index i10 71(2050 citations). In the bases Thomson Reuters ISI, Google Scholar, Scopus and Microsoft Academic. The total ~2800 citations. Conferences - Presentations and lectures delivered at more than 160 international congresses, conferences, symposia and seminars (Australia, Austria, Azerbaijan, Armenia, Belarus, Bulgaria, Canada, Chile, China, Czech Republic, Denmark, Estonia, France, Georgia, Germany, Great Britain, Greece, Hungary, Italy, Israel, Kyrgyzstan, Latvia, Poland, Russia, Serbia, Slovakia, Spain, Sweden, Switzerland, Turkey, Ukraine, USA, Uzbekistan), plenary lectures and membership in scientific and program committees of many international symposia.

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Last conferences with the invited lectures: 18th Int. Conf. on Computing in Engineering and Science (ICCES-2012) April 28 - May 5, 2012, Hania, Crete, Greece; 26th Symp. On Vibrations in Physical Systems+11th Int. Symp. on Trends in Continuum Physics May 3-9, 2014, Poznan, Poland; EUROMECH Colloquium 574 “Recent trends in modeling of moving loads on elastic structures” April 14-18, 2015, Eskišehir, Turkey; ChinaUkraine Sci. Forum of Science&Technology, July 3-11, 2016 HarbinHohhot, China; 7th Int. Conf. “Auxetics and other materials and models with “negative” characteristics” September 12-16, 2016 Gdansk, Poland; China (Dongguan) Int. Science and Technology Cooperation Week, December 7-10, 2017, Shandung, China; Western Returned Scholars Association Symposium, October 25-30, 2018, Zhengzhou, China; Global Summit on Nanotechnology, September 18-20, 2019, Barcelona, Spain. Cooperative scientific activities - Scientific missions within the framework of common projects (Poland, IPPT–1973, 1978, 1986, 1993), (DDR, Inst Mech, Dresden University, Magdeburg Techn. Hochschule, Chemnitz University etc-1986, 1988), (Czech Rep, UTM, UTAM – 1987,1989-1991), (Slovak Rep, IMM – 1989-1992), (Bulgaria, IMBM – 1988), (Hungary, Budapest Techn University – 1989), (Germany, Erlangen-Nürnberg University, 1994), (Italy, Universita di Roma “La Sapienza,” Universita di Salerno, 2004-2006,2008), (Scotland, University of Aberdeen, 2003-2010), (Turkey, Yieldiz University, 2011). Active cooperative work with Universita di Roma “La Sapienza” and Universita di Salerno (Italy) and the Centre for Micro and Nanomechanics at University of Aberdeen (Scotland). Advanced teaching activities - 12 candidate dissertations (PhD) Awards, prizes, honorary academic status, scientific grants 3 USSR government medals (1965, 1970, 1980); State Prize of Ukraine in Science, 2015; Award of Parliament of Ukraine, 2018; Medal of the National Academy of Sciences of Ukraine “For Professional Achievements”, 2008; Medal of the National Academy of Sciences of

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Ukraine “To Centenary of the Academy”, 2018; Dinnik A.N. Prize of the National Academy of Sciences of Ukraine, 2005; Glushkov V.M. Prize of the National Academy of Sciences of Ukraine, 2015; Honorary Professor, Engineering School, University of Aberdeen, Scotland, UK, 2011. Many tens of international scientific grants. From Soros and American Physical Society grants to the Royal Society of London and Royal Society of Edinburg grants. Memberships - ASME Int (American Society of Mechanical Engineers), 1998; GAMM (Gesellschaft für angewandte Mathematik und Mechanik), 1993, official representative of Ukraine; EUROMECH (European Mechanical Society), 1995; American Nano Society, 2011; New York Academy of Sciences, 1995; Ukrainian Physical Society, 1994; Ukrainian Society of Mechanical Engineers, 1993; National Committee of Ukraine of Theoretical and Applied Mechanics, 1993; Secretary of the National Committee of Ukraine of Theoretical and Applied Mechanics, 2000; Member of the Editorial Boards of international scientific journals: Int. Appl. Mech. (Kyiv-New York, Springer Group); Int. J. Appl. Math. and Mech. (Cardiff-Hong Kong, Research India Publications); Int. J. Mech. and Solids (New Mexico, Research India Publications); Nonlinear Dynamics and Systems Theory (Kyiv, InforMath Publishing Group); Waves, Wavelets, Fractals: Advanced Analysis (Rome-Warsaw, De Gruyter). Research Interests, Main Scientific Results - Mechanics of nano- and microcomposite materials, linear and nonlinear waves in materials. In the area of hereditary (viscoelastic) media: the series of works on applicability of the Volterra’s correspondence principle to the contact problems of viscoelasticity, on uniqueness and existence theorems is written; the textbook for universities is published. In the area of multiphase media (composite materials): the new scientific direction is developed– the microstructural theory of elastic, viscoelastic,

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and piezoelastic mixtures. Here a number of basic problems of statics, dynamics and stability is solved and the theory of elastic mixtures is constructed like the classical theory of elasticity. The microstructural and nonlinear physical phenomena and effects are detected and described. These results are included into four scientific monographs and into a few invited author’s reviews in the journal “International Applied Mechanics.” In the area of theory of waves: some new nonlinear models for waves in materials are proposed and analyzed, a number of new nonlinear wave effects in materials are described – new harmonics generation, wave evolution, triplets and quadruplets generation, energy pumping etc. A theoretical basis for the working on hyperesound waves nanotransistors is created. New waves are detected and new wave classification is proposed. The invited review in the best world review journal “Applied Mechanics Reviews” (USA) is published. In the area of applied theory of wavelets a number of textbooks jointly with professors of Kyiv Polytechnic Institute were published. The new notion of elastic wavelets is proposed and the way of application of elastic wavelets in problems of propagation of solitary elastic waves is shown. In the area of composite materials: the fundamental problems of micro and nano-mechanics of composite materials are investigated. On examples of standard fibrous and nonstandard (whiskerized microfibers or bristled nanofibers and nanogranules) fibrous and granular composites, the features of continuum structural models of mechanics are shown. The new structural models were developed as applied to micro- and nanolevel materials. In the area of auxetic materials: the new concept of auxetic phenomenon is proposed based on the nonlinear elastic models.

In: Understanding Plane Waves Editor: William A. Cooper

ISBN: 978-1-53616-779-5 c 2020 Nova Science Publishers, Inc.

Chapter 4

S PACETIME S YMMETRIES AND I NTERACTION OF Q UANTUM R ELATIVISTIC PARTICLES WITH E XTERNAL P LANE WAVE F IELDS H. K. Ould-Lahoucine∗ Laboratory of Mechanics and Advanced Materials National Polytechnic School of Constantine, Constantine, Algeria Dedicated to the memory of my Father

Abstract The subject of relativistic quantum particles interacting with classical plane wave fields is examined from the standpoint of the space-time symmetries which have been found to be encoded in the solutions of relativistic equations. Principally, it is shown how the elements of the proper Lorentz group come into play as a basic ingredient to get the solution of the Dirac equation under the form of variable transformations acting on the free-field solution. Subsequently, this underlying Lorentz structure is also found in the full solutions of spin  1 particles in interaction with classical plane wave fields by mean of a local gauge, a Lorentz and displacement transformations (ULT) acting as variable transformations on the free-field. On the other hand, considering the role ∗

Corresponding Author’s E-mail: [email protected]

204

H. K. Ould-Lahoucine played by the relativistic Green function as a fundamental object in the description of several scattering processes in quantum electrodynamics (QED) involved with the electromagnetic plane waves, an exhaustive review is done of the different approaches devoted to its derivation including the algebraic methods, path integrals and worldline formalisms.

Keywords: space-time symmetries, Dirac’s equation, Dirac’s particle, relativistic Green’s function, path integrals

1.

Introduction

The role played by the electron in several domains of physics still captivate more and more interest in the theorists physicists community as well as the
experimentalists one. Indeed, having an electric charge, a mass and a spin 12 , the electron interacts with the electromagnetic and gravitational fields and this fact led precisely Einstein to sum up this situation by saying: “Physics is the electron!”. In order to understand the importance of each property of the electron, and thus the Dirac equation, it is worth remembering that a complete explanation of the spin as an intrinsic property is still relevant since it has not been completely elucidated and remains, among all known observables, that for which a satisfactory physical interpretation is still lacking. Originally, this quantity was introduced to explain, among other questions, the incoherence of the atomic magnetism theory simply reduced at the beginning to the single orbital movement of electrons [1]. Since then, spin has not ceased to captivate an increasingly interest, not only in theoretical physics, but in all fields where this quantity is involved. An exhaustive summary of this question can be found in [2] and we only quote here the following facts: - In Chemistry: the fact that electrons obey Pauli’s principle is a crucial fact that conditions all chemistry. It is now well known that if the electron were a boson, ions with a significant negative charge would have existed, so that matter would have been thermodynamically unstable. - In Quantum Chromo Dynamics (QCD): it is unclear how the combination of quarks (spin− 21 particle) by mean of gluons leads to protons or neutrons which are also spin− 21 particles.

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It is undeniable that one of the most brilliant contributions of Dirac was the equation describing the dynamics of the electron in a quantum relativistic framework [3]. Besides having predicted the existence of the anti electron (positron), this equation was also at the origin of a major revision in physics, that of the notion of elementary particle. Indeed, just like the electron, the proton and the neutron are spin-1/2 fermions,   for which the predictions gave a magnetic e moment equal to µp = gp 2mp c , with gp ' 2, and µn = 0, respectively. However, Stern (1933), and later Rabi (1936), had found a surprising shift of three units for gp, while, for their part, Alvarez and Bloch (1940) had obtained gn = −3, 82629 for the neutron, where the sign (-) indicates that the direction of the magnetic moment is opposite to that of the spin. Since then, the term “elementary particle” for the proton and the neutron has become problematic. Thereafter, the discovery of the quarks has come to give a more refined desciption of the microscopic behavior of the matter, expected what would the string theory add to our understanding as the ultimate theory explaining the universe. The solutions of the Dirac equation for a charged particle interacting with an electromagnetic plane wave found by Volkov [4] have gained wide applications in several radiative processes of QED, for instance, the non linear Compton effect, pair creation or more recently with the development of high-power laser systems. As particular outcome, the mathematical properties of the Volkov states were the subject along the time to meaningful investigations that had focused on some aspects such as orthogonality, normalization or completeness, but it appeared from the beginning that they encode more deeper insights which are connected to the spacetime symmetries. Indeed, Taub [5] was the first who remarkably exhibited an underlying Lorentz structure of the Volkov’s solutions, much after which Kupersztych [6] shown that the wave function for a Dirac’s particle interacting with a classical plane wave field is nothing but a variable transformation of the proper Lorentz group acting on the free-field solution, and later on, Brown and Kowalski [7] succeeded also to include for the same case the full solutions of spin ≤ 1 particles by mean of a local gauge, a Lorentz and displacement transformations (ULT). This chapter aims to emphasize these aspects to complete our understanding of the properties of the plane waves and to lead researchers, hopefully, to be more interested in this question which seems to be fundamental. Of course, the chronology of these developments is followed, starting with recalling in the first part the solutions of the Dirac equation in the presence of a plane wave, first found by Volkov. In the second part, the connection of the Volkov solutions

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to the spacetime symmetries is mentioned, while in the third part an overview of the different formulations of the path integrals for the Dirac particle interacting with a plane wave field is given, since they are perceived as an equivalent formalism to solving the Dirac equation.

2.

The Solutions of The Dirac Equation in a Plane Wave Field: The Volkov’s Solutions

Historically speaking, Dirac’s equation came in response to some difficulties encountered with the second-order Klein-Gordon equation that turned out to be useful for describing the dynamics of spin-zero particles only, while the electron is a spin− 12 particle, this in addition to the question of interpreting the negative energy solutions as well as the probability density. Setting the goal, from the beginning, to get a quantum relativistic first order equation, Dirac succeeded to → show that, If the four components vector Φ (− r , t) represents the wave function that encodes the full information about the motion of an electron of mass m and → charge e, then Φ (− r , t) must verify → [iγ µ ∂µ − m] Φ (− r , t) = 0,

(1)

where γ µ are the 4x4 Dirac matrices and ∂µ stands for the derivation with respect to the spacetime coordinates. In presence of an external field Aµ , the Dirac equation then takes the form → [γ µ (i∂µ − eAµ ) − m] Ψ (− r , t) = 0

(2)

and we then speak about coupling between the particle and the external field, that is the most important configuration in physics since the notion of “free particle” is an idealization rather than a palpable reality. It soon became apparent that this equation effectively overcame the difficulties encountered with that of Klein-Gordon, which has led many scientists to seek solutions for certain forms of interactions as it has done by Volkov (see [4]) with an electromagnetic plane wave. As a reminder, the case of the electromagnetic plane wave Aµ (φ) is characterized by the quantity φ = k.x = kµ xµ , kµ being the wave vector and xµ the position four vector. Also, by assigning a null mass to the photon at rest (as required by special relativity), it follows that:

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207

k2 = kµ kµ = 0, while the Lorentz gauge is written d (k.A) =0 k.x which implies that k.A = cte. For the sake of simplicity, this constant can be set for zero k.A = 0, ∂µ Aµ =

meaning that the wave vector kµ is transverse to the external electromagnetic field Aµ , which is effectively the case in the vacuum. Without spreading out all the details of the calculation to solve Eq. (2), we just notice that sometimes it is better to switch to the quadratic form by multiplying this equation by [γ µ (i∂µ − eAµ ) + m]. Therefore, using the properties of the Dirac matrices one gets h   i 0 → −∂ 2 + e2 A2 − 2ie (A.∂) − (γ.k) γ.A − m2 Ψ (− r , t) = 0, 0

where the notation γ µ aµ = γ.a is used and Aµ stands for the derivative of Aµ with respect to φ. A useful method to solve this kind of equations is to opt for a separation of variables assuming that → Ψ (− r , t) = e−ip.x F (φ) so that we get after replacing in the equation above h  i 0 0 2i (k.p) F + −2e (A.p) + e2 A2 − ie (γ.k) γ.A F = 0, 00

understood that ∂µ ∂ µ F = k2 F = 0. The integration of the previous differential equation is obvious and after some arrangements one gets 

    Z k.x  e u e2 2 i F (φ) = 1 + (γ.k) (γ.A) e (A.p) − A dφ exp − 2k.p k.p 0 2 (2p0 )1/2

where

u (2p0 )1/2

is an arbitrary bispinor. Finally, the solution sought is

 − → Ψ ( r , t) = 1 +

 e u (γ.k) (γ.A) exp (−iS) 2k.p (2p0 )1/2

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with 1 S = p.x + k.p

Z

0

k.x



 e2 2 e (A.p) − A dφ. 2

the last two expressions constitute Volkov’s solution to the Dirac equation found on 1935. Ever since this pioneering work, these solutions give rise continuously to wide interest and the number of papers that continues to appear in relation with this subject is so large that it would be impossible to quote them here.

3.

The Volkov’s Wave and the Connection to the Spacetime Symmetries

Historically, it seems that the first research devoted to the existing connections between space-time symmetries and the external fields of plane electromagnetic waves is related to Taub [8] who had showed that the solutions of classical relativistic equations of motion for an interacting charged particle with a plane electromagnetic wave can be derived by mean of Lorentz matrices. Later, Taub (see [5]) also managed to show that the solutions of the Dirac equation for an electromagnetic plane wave field are obtained by transforming solutions for a free electron by Lorentz matrices. However, not all forms of interaction may have this aspect, because Taub also established the necessary and sufficient conditions that must be satisfied. Indeed, we must have: pν ∂σ l νµ xµ + ∂σ S − eAσ = 0 and γ µ ∂µ T = 0, where pν is a constant four vector, Aσ is the external field, l νµ are the elements of the inverse matrix of the Lorentz matrix £µν verifying l νµ £µρ = δ νρ where δ νρ is the Kronecker delta, S is an arbitrary function of the coordinates, γ µ are the Dirac matrices while T is the spin operator transformation. As mentioned also by Taub in [5], a constant magnetic field with a given direction does not satisfy the conditions expressed above which shows that the connection to the spacetime symmetries is a property of fields having the structure of a plane wave.

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Later, Kupersztych (see [6]) succeeded to obtain a result similar to that of Taub throughout a innovating approach that combines gauge invariance and relativistic invariance. As pointed out by Schwinger [9], relativistic and gauge invariance have a prominent role among the invariance properties that characterize Quantum Electrodynamics (QED) and this fact was especially enlightened in the work of Kupersztych for a quantum relativistic particle. Indeed, the basic ingredients in this approach are the symmetry properties of the free field and gauge invariance, both brought into play to give the desired result. Briefly, the starting point in the work of Kupersztych are the gauge invariance of the Dirac equation in presence of an external field and the free-field Dirac equation, respectively: h  h i i b − m eieΛb Ψ = 0 γ µ i∂µ + eAµ + e ∂µ , Λ (3) and

[iγ ν ∂ν − m] Φ = 0,

(4)

where the wave function Φ is solution to the free-field Dirac equation (4) while (3) traduces the invariance of hthe equation (2) under the generalized gauge i µ µ b b depend on transformations A → A + ∂, Λ when the gauge functions Λ the operators ∂µ and xµ . The matrix elements £µν of the Lorentz transformation £ are connected to the spin variable operator Tb via the condition £µν γ ν = Tb−1 γ µ Tb

and consequently, one can re-express (4) under the form h i  µ ρ  iγ £ µ ∂ρ − m TbΦ = γ µ i£ρµ ∂ρ ,Tb Φ.

(5)

(6)

Obviously, this approach could h notiwork for an arbitrary external field exµ b b and £ commute. Thus, concretely, the cept if the operators A (ϕ), Λ, ∂ µ , Λ equations (3) and (6) equivalent if one has simultaneously h i b i£ρµ ∂ρ = i∂µ + eAµ + e ∂µ , Λ (7) and

h i γ µ i£ρµ ∂ρ ,Tb = 0

(8)

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so that, by using the properties of the elements of the Lorentz group, one arrives at   d b ν ν µ ν µ ν µ ,Λ ∂ . (9) i ∂ν ∂ = i£ ν ∂µ ∂ + e£ ν Aµ ∂ + e£ ν kµ dϕ

The quantities of the right hand side of (9) can be computed from (7) by contraction with ∂ µ , Aµ and kµ , respectively, which leads after integration to b under the form the expression of Λ   ϕ  Z Zϕ 1 b = −  A (η) dη  .∂ + ie A2 (η) dη  (k .∂)−1 Λ (10) 2 0

0

by pointing out that the action of the operator (k .∂)−1 on the plane wave basis cannot be indefinite since the electron is a non zero mass particle. Turning next to the question of the spinor operator T , it should be reminded first that the general methods of spinor calculus tell us that the spinor operator which corresponds to Lorentz-type operator can be constructed as the product of the spinor operator corresponding to a Lorentz transformation £ (without rotation) and the spinor operator corresponding to a rotation operator R Tb = S´ (R) S´ (£)

The detailed calculus can be found in [6] for the interested reader and we simply limit ourselves to give the expression of the spinor operator sought which is e Tb = 1 + (k .∂)−1 (γ.k) (γ.A) 2 b are now identified, the equations (3) and and hence, since the operators Tb and Λ (6) allow finally to write the wave function for a quantum relativistic electron interacting with an electromagnetic plane wave field under the form   b Φ Ψ = Tb exp −ieΛ

The previous expression traduces remarkably the role played by the symmetries of the free-field as well as gauge invariance as basic ingredients to get the final result. Until then, this underlying Lorentz structure was found both for the relativistic equations of motion and for the wave functions of the spin− 12 particles,

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211

but after the work of Brown and Kowalski (see [7]), it has been found that the symmetries associated with the plane waves can be generalized so that they now concern all the particles having
a spin ≤ 1. Firstly, we recall that the wave equations for the three spins 0, 21 , 1 are given by
D 2 + m2 Ψ = 0 (scalar); (11) (iγ.D − m) Ψ = 0 

(Dirac);


D 2 + m2 Ψµ + 2iQFµν Ψν = 0, D.Ψ = 0 (vector),

(12)

(13)

with Dµ = ∂µ + iQAµ (Q being the charge) and Fµν the electromagnetic field strength tensor. The full solutions to the previous equations were found to be of the generic form Ψ(x) = U LT Φ(x) where Φ(x) is the free solution and U LT represents the product of local gauge (U ), Lorentz (L) and displacement (T ) transformations, respectively. Using a more compact notation, one can put the free solutions for the three spins under the form Φ(x) = e−ip.x {1; w(p); η (p)} such that the four momentum pµ verifies p2 = m2 , (γ µ pµ − m) w(p) = 0 and η (p) .p = 0. The transformations are given by L = eS = {1; 1 + F ; Λ} with

  Q S = 0; (γ.k) (γ.A) ≡ F ; (kµ Aν − kν Aµ ) ≡ Ωµν 2k.p

and Λµν = gµν + Ωµν −

Q2 A2 kµ kν . 2 (k.p)2

Also, we have U (θ) = eiQθ , θ =

Q k.p

Z

k.x

dzA2 (z)

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−ip.d

T (d) = e

Q ,d = k.p µ

Z

k.x

dzAµ (z) .

It should be noted that this generalization works only for the minimal derivative couplings and is no longer valid for instance with anomalous magnetic moments.

4.

The Volkov’s Wave in the Framework of Path Integrals

The description of the electron in a quantum relativistic framework is not restricted solely to solving the Dirac equation, but can be done also by mean of the Feynman propagator (or the Green’s function) derived following several formalisms including path integrals and algebraic methods. In this context, it is well known that Schwinger (see [9]) was the first to calculate the Green’s function using a purely algebraic technique, followed then by Nikishov and Ritus [10], even though the results as well as the techniques were quiet different. Besides these algebraic techniques, one can find several formulations basically founded on functional integrations, commonly called path integrals, where the dynamics of the electron is described by the couple (x, p) of the habitual phase space and completed by additional variables to account for the spin degree of freedom. Even though this question was a long-standing problem first underlined by Feynman and Hibbs [11] who emphasized the ambiguity of representing a discrete quantity, namely the spin, by continuous variables, we can however say that this difficulty has been effectively overcoming throughout the supersymmetric action proposed by Berezin and Marinov [12] where fermionic (non commuting) and bosonic (commuting) variables are used. Later, Fradkin and Gitman [13] gave the path integral formulation for the Green’s function such

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213

that it reads in the case of the plane wave as follows Z ∞ Z  Z ∂l de0 dχ0 DxDeDpe DχDpχ DψM (e) Se c (x1 , x0 ) = exp iγ n n ∂θ 0   2 ·  Z 1 · − x − e m2 − g xA (k.x) + iegFµν (k.x) ψ µ ψ ν exp i  0 2e 2 ! # · ·n xα ψ α · · 5 − mψ χ − iψ n ψ + pχ χ + pe e dτ +i e +ψ n (1) ψ n (0)} |θ=0 .

In this formulation x, e and pe are bosonic (commuting) variables, whereas, the quantities θ, χ, pχ and ψ are Grassmannian (anticommuting) variables with the following initial conditions: x (0) = xa, x (1) = xb , e (0) = e0 , χ (0) = χ0 , ψ (1) + ψ (0) = θ . An interested reader can find, for instance in [14], the detailed derivation of the green’s function after successive integrations over bosonic and fermionic variables such that +∞ Z

d4 p exp [ip. (xb − xa)] (2π)4 −∞     e e −1 / / k/A(k.xb ) (γ.p − m) 1− k/A(k.xa) 1+ 2k.p 2k.p  k.x  Zb i 1 2 2 exp  ep.A − e A dη k.p 2

G (xb , xa) = −

k.xa

i − (k.p)2

k.x Zb

k.xa

   1 2 2 ep.A − e A dη  . 2

Another path integral formulation of the Dirac electron was also given by Barut and Zanghi in [15] where the spin degree of freedom is represented by bosonic variables only. The Green’s function takes the from  Z ∞  τ  i G (xb , xa) = − (14) dτ exp −i m Gτ (xb , xa) , λ λ 0

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H. K. Ould-Lahoucine

such that  λ τb G (xb , xa) = Dp Dx Dz Dz exp − (z z) |τ a 2  Z τ     ·
λi · · / exp i dτ p.x − z /p − eA z + z z − zz , 2 0 τ

Z



Z

where τ = τ b − τ a is the time transition and λ = ±1, respectively for the particle and the antiparticle while the dots stand for the derivative with respect to the time transition τ . The procedure in this case is practically the same as that of Fradkin and Gitman () which consists in dividing the time transition in (n + 1) subintervals with length  : τ = τ b − τ a = (n + 1) , such that



xb = xn+1, xa = x0 .

This procedure is called discretization and leads to write the function Gτ (xb , xa) under the form Z n+1 n Y d 4 pj Z Y X Z +∞ n+1 Y  iλ  4 G (xn+1 , x0 ) = lim d xj dz jσ dzρj 4 n→∞ 2π (2π) σ,ρ −∞ j=1 j=1 j=1 ε→0   n+1 X exp i [pj . (xj − xj−1 )  τ

j=1


λ j z σ δ σρ zρj − zρj−1 i  −ε z jσ (γ.pj − eγ.A (k.xj )) zρj ,

+

followed then, theoretically, by successive integrations over the full variables, and after that, the use of (14) gives finally the desired result. However, the practical calculus are not so obvious since it is known that, in path integrals, we can obtain only for the free case (A = 0) an exact result and the use of the perturbation theory is sometimes systematic. A possible alternative to the use of the perturbation theory can be found in [16, 17, 18] in the case of a classical plane

Spacetime Symmetries and Interaction of Quantum Relativistic ...

215

wave field Aµ (k.x) where the previous expression is remodeled by bringing into play the transformations T and T −1 such that we have now for the function Gτ (xb , xa) :  λ τb G (xb , xa) = Dp Dx Dz σ Dzρ exp − (z σ δσρ zρ ) |τ a 2 σ,ρ   Z τ    ·
λi · · z σ δ σρ zρ − z σ δ σρ z ρ × Tb−1 βσ exp i dτ p.x + 2 0
−1  −z σ T γ σρ . p − e γ σρ .A (k.x) T zρ (Ta)ρα . τ

Z

XZ



This technique, first due to Duru and Kleinert [19] as a regularization method of singular potentials, was thus generalized so that the regularization functions T and T −1 are now a combination of Dirac matrices and having the role of reconverting the problem of path integrals with an external field to that of the free field. It should be noted that the previous formulation works only for external fields having the plane wave structure since the transformations T and T −1 are so that they depend generally on the operators kµ x bµ and kν pbν which commute [kµ x bµ , kν pbν ] = 0,because of the property kµ kµ = 0. Consequently, the determination of the transformations T and T −1 allows one to derive straightforwardly the expression of the propagator after successive integrations.

Summary The symmetries of the spacetime are intimately connected to the solution of the Dirac equation in presence of an electromagnetic plane wave field (Volkov’s solution). The latter is a fundamental tool in investigating several processes in Quantum Electrodynamics (QED) by regarding the role played by the electron as the seminal object in the full radiation-matter interactions. Consequently, investigating such connections could led to better understanding the deep insights that are encoded in these processes and which are likely to give us a more complete vision of the radiation-matter interaction. This point of view also deserves to be generalized in the case of Green’s function with regard to the role it plays in QED, where it is seen sometimes as the most important object.

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References [1] G. E. Uhlenbeck and S. Goudsmit, Naturwiss, 13, 953 (1925). [2] J. Fr¨ohlich, ”Spin, or actually: Spin and Quantum Statistics”, Progress in Mathematical Physics, 55 (2009). [3] P. A. M. Dirac, Proc. R. Soc. Lond, A 117, 610-624 (1928). [4] D. M. Volkov, Zh. Eksp. Teor. Fiz. 7, 1286 (1939). [5] A. H. Taub, Rev. Mod. Phys. 21, 388 (1949). [6] J. Kupersztych, Phys. Rev. D 17, 629 (1978). [7] R. W. Brown, and K. L. Kowalski, Phys. Rev. Lett, 51, 2355 (1983). [8] A. H. Taub, Phys. Rev. 73, 786 (1948). [9] J. Schwinger, Phys. Rev., 82, 644 (1951). [10] A. I. Nikishov and V. I. Ritus, Sov. Phys. JETP 19, 529 (1964). [11] R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals, (McGraw-Hill, New York, 1965). [12] F. A. Berezin and M. S. Marinov, Ann. Phys. 104, 366, (1977). [13] E. S. Fradkin and D. M. Gitman, Phys. Rev. D 44, 3230, (1991). [14] S. Zeggari et al., Czec J Phys, 51(3), 185-198 (2001). [15] A. O. Barut and N. Zanghi, Phys. Rev. Lett. 52, 2009 (1984). [16] H. K. Ould-Lahoucine and L. Chetouani, Cent. Eur. J. Phys. 7, 184-192 (2009). [17] H. K. Ould-Lahoucine and L. Chetouani, Int. Jour. Theo. Phys. (2012). [18] H. K. Ould-Lahoucine and L. Chetouani, J. Math. Phys, 53, 072303 (2012). [19] H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics and Financial Markets, 3rd edition, World Scientific Publishing, Singapore (2004).

INDEX

A acoustic Goos-Hänchen effect, vii, viii, 2, 5, 16 acoustic measurement, v, 49, 105, 112, 113, 114, 139, 140, 141 acoustic network, 114 acoustical measurements, 2, 114, 141 acoustical-logging, 2 acoustics, vii, 1, 3, 4, 5, 105, 106 amplitude, 3, 6, 15, 46, 49, 51, 52, 53, 55, 56, 57, 58, 60, 69, 73, 74, 75, 76, 77, 102, 105, 106, 126, 127, 128, 132, 134, 135, 138, 140, 153, 155, 158, 159, 161, 163, 164, 166, 168, 171, 173, 174, 175, 176, 180, 181, 182, 184, 188, 190, 192 anisotropic media and interface, 2 anisotropy, viii, 2, 3, 4, 29, 30, 43, 46, 48, 60, 78, 79, 82, 83, 84, 85, 86, 87, 88, 102, 103, 105, 107, 110, 111, 112, 144 anomalous incident angle, 2, 23, 25, 28, 105

B boreholes, 110 boson, 204

C convolution, 53, 106, 114, 127 crystals, 3, 29, 31 cubic system, 151

D damping, 50, 55, 122, 123, 124, 125 depth, viii, 2, 79, 82, 83, 84, 85, 86, 87, 88, 110 dielectric constant, 118 Dirac equation, ix, 203, 204, 205, 206, 208, 209, 212, 215 discretization, 214 displacement, vii, viii, ix, 4, 5, 8, 9, 12, 13, 14, 15, 16, 26, 30, 61, 79, 82, 92, 99, 103, 106, 108, 114, 118, 136, 142, 147,

218

Index

149, 159, 160, 162, 163, 166, 167, 169, 176, 186, 189, 190, 191, 192, 197, 203, 205, 211 distortion of initial wave profile, 148

E electromagnetic, ix, 2, 204, 205, 206, 207, 208, 210, 211, 215 electromagnetic waves, 2 electron, 204, 205, 206, 208, 210, 212, 213, 215 energy, 3, 29, 30, 31, 32, 72, 73, 75, 77, 78, 79, 80, 81, 82, 83, 85, 102, 105, 154, 155, 167, 201, 206 energy conservation, 102, 105 energy density, 102 evolution, vii, viii, 147, 148, 149, 150, 151, 152, 154, 155, 157, 159, 160, 161, 162, 163, 164, 165, 166, 168, 170, 176, 178, 179, 180, 183, 189, 190, 191, 192, 201 excitation, viii, 57, 73, 75, 77, 105, 113, 126, 135, 136

F fluid, 3, 4, 5, 6, 12, 14, 15, 16, 53, 54, 56, 66, 67, 69, 70, 71, 73, 74, 105, 106, 109, 110, 115 force, 59, 116, 117, 119, 120 formation, 14, 49, 56, 57, 59, 60, 64, 66, 70, 73, 74, 75, 76, 77, 106, 114, 174, 178

G gated sine-wave, 114, 135, 136 geometrical parameters, 4, 50, 51, 68, 139, 140 gluons, 204

I integration, 139, 207, 210 interface, vii, viii, 2, 3, 4, 5, 6, 7, 9, 12, 13, 14, 15, 16, 20, 21, 25, 26, 37, 39, 40, 42, 45, 46, 47, 48, 49, 58, 59, 60, 61, 89, 91, 92, 93, 97, 100, 101, 102, 103, 105, 106, 108, 109, 111, 115, 130, 131, 132, 133, 135, 139, 143, 144 interference, 5, 106 inversion, 64, 115, 127, 139 ions, 204 isotropic media, 60, 78, 79, 107, 108

K Klein-Gordon equation, 206

L linear model, 151 linear superposition, 114, 115, 127 logging, vii, viii, 2, 4, 5, 8, 13, 14, 49, 53, 54, 55, 56, 57, 58, 60, 64, 66, 68, 69, 70, 71, 72, 73, 75, 76, 105, 106, 110, 112, 142, 144 longitudinal hyperelastic wave, 148

M magnetic moment, 205 magnetism, 204 magnitude, 79, 80, 103 matrix, 3, 18, 61, 79, 88, 89, 90, 93, 94, 99, 100, 208, 209 measurement network, 2, 105, 106, 128, 138 mechanical properties, 89, 142, 184, 192 media, vii, 1, 2, 3, 4, 17, 20, 29, 37, 41, 46, 47, 53, 54, 59, 61, 76, 78, 80, 89, 91, 92,

Index 93, 97, 99, 100, 102, 105, 106, 107, 108, 114, 115, 140, 197, 200 Murnaghan’s model (Murnaghan’s nonlinear model), 148, 149, 186, 192

N nanocomposites, 195, 197 nanofibers, 201 nanomaterials, 197 natural hazards, 114 neutrons, 204 numerical analysis, 165, 189

219 Q

quantum electrodynamics (QED), ix, 204, 205, 209, 215 quarks, 204, 205

R radiation, 15, 72, 73, 74, 75, 110, 116, 125, 126, 130, 139, 140, 215 reflection and refraction, 2 reflectivity, 64, 66 relativistic Green’s function, 204 resistance, 116, 119, 120, 125, 126 restrictions, 162, 164, 176, 192, 197

O ordinary differential equations, 183

P path integrals, ix, 204, 206, 212, 214, 215 piezoelectric material PZT4, 114, 136 plane wave refection/refraction, 114 plane waves, ix, 148, 149, 192, 204, 205, 211 polarization, viii, 2, 3, 4, 21, 22, 25, 26, 28, 29, 30, 31, 32, 37, 39, 40, 41, 42, 43, 46, 47, 48, 49, 79, 91, 92, 93, 95, 96, 97, 99, 105, 111, 112, 114, 144, 145 polarization states, viii, 2, 4, 29, 37, 44, 46, 47, 105 propagation, viii, ix, 2, 3, 8, 12, 13, 14, 15, 16, 30, 32, 44, 49, 53, 54, 55, 59, 67, 80, 85, 105, 106, 109, 110, 113, 115, 126, 134, 135, 136, 139, 140, 147, 149, 159, 160, 162, 163, 164, 166, 167, 169, 182, 184, 190, 191, 201

S scattering, ix, 4, 5, 59, 105, 204 seismic data, 3, 78, 79, 85, 88, 105 seismic signal and analysis, 2 signals, 53, 57, 58, 60, 69, 71, 73, 76, 79, 88, 105, 106, 114, 115, 126, 130, 134, 137, 138, 142 sine wave, 132, 133 spin, ix, 203, 204, 205, 206, 208, 209, 210, 211, 212, 213 successive approximations, vii, viii, 147, 158, 176, 181, 188, 192 symmetry, 3, 17, 29, 60, 78, 88, 110, 119, 209

T transmission, viii, 14, 16, 17, 53, 54, 55, 60, 61, 63, 67, 68, 69, 70, 107, 108, 110, 113, 114, 115, 117, 125, 126, 127, 135, 139, 140, 142 tunneling, 109

220

Index U

ultrasound, 114, 143, 159

V variables, 55, 207, 212, 213, 214 variants of approximate analysis of wave evolution, 148 vector, 26, 28, 29, 30, 31, 62, 79, 80, 81, 152, 206, 208, 211 velocity, 20, 21, 24, 25, 26, 30, 31, 32, 49, 57, 58, 59, 66, 69, 73, 75, 76, 77, 78, 79, 80, 81, 82, 83, 88, 93, 94, 95, 97, 106, 116, 120, 126, 136, 151, 155, 156, 162, 164, 174, 185, 186, 187, 190 vibration, 92, 116, 119, 120, 125, 126

W wave number, 15, 154, 155, 158, 163, 183, 184 wave propagation, ix, 7, 105, 108, 148, 151, 157, 159, 162, 163, 166, 168, 182, 188, 190, 192, 194 wave vector, 29, 31, 81, 206, 207 wavelet, 5, 6, 8, 14, 15, 49, 54, 55, 56, 64, 66, 80, 106, 109, 115, 125, 127, 196 wavelet analysis, 196