WAVES IN FLUIDS AND SOLIDS Edited by Rubén Picó Vila Waves in Fluids and Solids Edited by Rubén Picó Vila Published by InTech Janeza Trdine 9, 51000 Rijeka, Croatia Copyright © 2011 InTech All chapters are Open Access articles distributed under the Creative Commons Non Commercial Share Alike Attribution 3.0 license, which permits to copy, distribute, transmit, and adapt the work in any medium, so long as the original work is properly cited. After this work has been published by InTech, authors have the right to republish it, in whole or part, in any publication of which they are the author, and to make other personal use of the work. Any republication, referencing or personal use of the work must explicitly identify the original source. Statements and opinions expressed in the chapters are these of the individual contributors and not necessarily those of the editors or publisher. No responsibility is accepted for the accuracy of information contained in the published articles. The publisher assumes no responsibility for any damage or injury to persons or property arising out of the use of any materials, instructions, methods or ideas contained in the book. Publishing Process Manager Sandra Bakic Technical Editor Teodora Smiljanic Cover Designer Jan Hyrat Image Copyright Alexey Chechulin, 2011. Used under license from Shutterstock.com First published September, 2011 Printed in Croatia A free online edition of this book is available at www.intechopen.com Additional hard copies can be obtained from orders@intechweb.org Waves in Fluids and Solids, Edited by Rubén Picó Vila p. cm. ISBN 978-953-307-285-2 free online editions of InTech Books and Journals can be found at www.intechopen.com Contents Preface IX Part 1 Elastic Waves in Solids 1 Chapter 1 Acoustic Waves in Layered Media - From Theory to Seismic Applications 3 Alexey Stovas and Yury Roganov Chapter 2 Soliton-Like Lamb Waves in Layered Media 53 I. Djeran-Maigre and S. V. Kuznetsov Chapter 3 Surface and Bulk Acoustic Waves in Multilayer Structures 69 V. I. Cherednick and M. Y. Dvoesherstov Chapter 4 The Features of Low Frequency Atomic Vibrations and Propagation of Acoustic Waves in Heterogeneous Systems 103 Alexander Feher, Eugen Syrkin, Sergey Feodosyev, Igor Gospodarev, Elena Manzhelii, Alexander Kotlar and Kirill Kravchenko Chapter 5 Multiple Scattering of Elastic Waves in Granular Media: Theory and Experiments 127 Leonardo Trujillo, Franklin Peniche and Xiaoping Jia Chapter 6 Interface Waves 153 Hefeng Dong and Jens M. Hovem Chapter 7 Acoustic Properties of the Globular Photonic Crystals 177 N. F. Bunkin and V. S. Gorelik Part 2 Acoustic Waves in Fluids 209 Chapter 8 A Fourth-Order Runge-Kutta Method with Low Numerical Dispersion for Simulating 3D Wave Propagation 211 Dinghui Yang, Xiao Ma, Shan Chen and Meixia Wang VI Contents Chapter 9 Studies on the Interaction Between an Acoustic Wave and Levitated Microparticles 241 Ovidiu S. Stoican Chapter 10 Acoustic Waves in Bubbly Soft Media 257 Bin Liang, Ying Yuan, Xin-ye Zou and Jian-chun Cheng Chapter 11 Inverse Scattering in the Low-Frequency Region by Using Acoustic Point Sources 293 Nikolaos L. Tsitsas Preface Acoustics is a discipline that deals with many types of fields wave phenomena. Originally the field of Acoustics was consecrated to the sound, that is, the study of small pressure waves in air detected by the human ear. The scope of this field of physics has been extended to higher and lower frequencies and to higher intensity levels. Moreover, structural vibrations are also included in acoustics as a wave phenomena produced by elastic waves. This book is focused on acoustic waves in fluid media and elastic perturbations in heterogeneous media. Acoustic wave propagation in layered media is very important topic for many practical applications including medicine, optics and applied geophysics. The key parameter controlling all effects in layered media is the scaling factor given by the ratio between the wavelength and the layer thickness. Existing theory mostly covers the solutions derived for the low-frequency and high-frequency limits. In practice, the wavelength could be comparable with the layer thickness, and application of both frequency limits is no longer valid. The frequency-dependent effects for acoustic waves propagating through the layered media are analyzed. Solitons, or by the original terminology “waves of translation”, are a special kind of hydrodynamic waves that can arise and propagate in narrow channels as solitary waves, resembling propagation of the wave front of shock waves. These waves can propagate without considerable attenuation, or change of form; or diminution of their speed. Motion of these waves can be described by a non-linear KdV differential equation. Soliton-like lamb waves are analyzed in the long-wave limits of Lamb waves propagating in elastic anisotropic plates. The application of various layers on a piezoelectric substrate is a way of improving the parameters of propagating electroacoustic waves. For example, a metal film of certain thickness may provide the thermal stability of the wave for substrate cuts, corresponding to a high electromechanical coupling coefficient. The overlayer can vary the wave propagation velocity and, hence, the operating frequency of a device. The effect of the environment (gas or liquid) on the properties of the wave in the layered structure is used in sensors. The layer may protect the piezoelectric substrate against undesired external impacts. Multilayer compositions allow to reduce a velocity dispersion, which is observed in single-layer structures. In multilayer film bulk acoustic wave resonators (FBAR) many layers are necessary for proper work of such devices. Wave propagation characteristics in multilayer structures are analyzed by means of general methods of numerical calculations of the surface and bulk acoustic wave parameters in arbitrary multilayer structures. Crystalline and disordered systems are analyzed as very peculiar systems. The most important elementary excitations appearing in them are acoustic phonons, which characterize vibration states in heterogeneous structures. In such systems, the crystalline regularity in the arrangement of atoms is either absent or its effect on the physical properties of the systems is weak, affecting substantially the local spectral functions of different atoms forming this structure. Granular materials consist of a collection of discrete macroscopic solid particles interacting via repulsive contact forces. Classical examples are sand, powders, sugar, salt and gravel, which range from tens of micrometers to the macroscopic scale. Their physical behavior involves complex nonlinear phenomena, such as non equilibrium configurati. The elastic wave propagation in confined granular systems under external load is developed from both experimental and theoretical viewpoints. Shear waves (S-wave) are essential in the field of seafloor geotechnical applications as they propagate in solids. More specifically interface waves and the use of the interface waves are important to estimate shear wave speed in the sediments as it provides a good indicator of sediment rigidity, as well as for sediment characterization, seismic exploration, and geohazard assessment. In addition, for environments with high seabed S-wave speeds, S-wave conversion from the compressional wave (also called P- wave) at the seafloor can represent an important ocean acoustic loss mechanism which must be accounted for in propagation modeling and sonar performance predictions. Phononic Crystals are characterized by spatial periodic modulations of the sound velocity caused by the presence of the periodically settled elements of various materials (metals, polymers etc.) inside the sample. The properties of acoustic waves in Phononic Crystals are in many respects similar to the properties of electromagnetic waves in Photonic Crystals. Periodic media can be characterized by the dispersion dependences ω(k) for acoustic waves together with the dispersion dependences of their group velocities and effective mass of the corresponding acoustic phonons. The results of the theoretical analysis and the data of experimental studies of the optical and acoustic phenomena in PTC and PNC, including the studies of spectra of non- elastic scattering of light together with the experiments to observe the stimulated light scattering accompanying by the coherent oscillations of globules are reported. The numerical solutions of the acoustic-wave equation via finite-differences, finite- elements, and other related numerical techniques are valuable tools for the simulation of wave propagation. These modeling techniques for the 1D and 2D cases are typically used as support for a sound interpretation when dealing with complex geology, or as a benchmark for testing processing algorithms, or used in more or less automatic [...]... Q N 1 11 Q 21   u 0  N 1 21 Q 22   d 0    (18 ) 8 Waves in Fluids and Solids From equation (18 ), it follows that the amplitudes of waves going away from the stack of layers: u 0 and d N 1 can be computed from the amplitudes of waves coming to the stack of layers: u N 1 and d 0 , d   T  u   R    N 1 D 0 D  d  u  ,   RU TU 0 (19 ) N 1 where 1 TD  Q 22  Q 21Q 11 Q12 ,... E 1   11 k ) (  iP 21 ( iP12k )  , (k )  P 11  (38) (k  (k  where Pmm)  E m cos  hΔ  Em1 and Pmn)  E m sin  hΔ  E n 1 , m  n, m, n  1, 2 From equation (37) and (38), it follows that  P( k ) Pk 1  exp  ihM k   E exp  ihΛ  E 1   11 ( k )  iP 21 ( iP12k )  , (k )  P 11  (39) and (  (k P11k )   E1 cos  hΔ  E1 1   E2 cos  hΔ  E 1  P22 ) 2 T T   (k T (k P 21 )... D)det C  2  c11c 21  c12c22   2 2  c11c 21  c12c22    2 2 2 2 2   det C   1  c 11  c12  c 21  c22  (57) where cij are elements of matrix C and 2 2 2 2 det  C  D  det C   det C   1  c 11  c12  c 21  c22 2 (58) The matrices RU and TU can be computed by interchanging the superscripts 1 and 2 or using the symmetry relations (37) and (56) (Ursin and Stovas, 20 01) , which gives T ... respectively 1 2 To approximate the reflection and transmission matrices in equation (54), we proceed as in j Stovas and Ursin (20 01) and expand the matrices Ek  , k  1, 2, j  1, 2 , into second-order Taylor series with respect to the average medium This gives the second-order approximations  2 TD 1 2 1    2 2 f  g12  g 11  g 22   g 11  g12 f  1  2  g 11  g12  f   2 2 , RD    1 1 2... (k P 21 )   E 2 sin  hΔ  E1 1   E 2 sin  hΔ  E1 1  P 21 ) T ( k )T P12   E1 sin  hΔ  E  1 T 2 (  E1 sin  hΔ  E 1  P12k ) 2 (40) Acoustic Waves in Layered Media - From Theory to Seismic Applications 11 The product of matrices of this type indicates that the propagator matrix P for the stack of layers can be blocked as follows iP12  P P   11 , iP 21 P22   ( 41) where the elements... that F and G have the form  0 F  f f  g 11 , G   0  g12 g12   g 22  (48) To compute the scattering matrices F and G , we express them as functions of perturbations in the elements of the matrices A and B in equation ( 31) The eigenvector matrices satisfy the following equations AE 2  E1Δ, BE1  E 2Δ From equations (35) and (36), we obtain (49) 12 Waves in Fluids and Solids  E1  E1 ... the vertical slownesses for qP- and qSV -waves,  qP and  qSV The transformation matrix E 1  E1  2  E2 E1   E 2  (35) is normalized with respect to the vertical energy flux so that the inverse has the simple form E 1    1  E1 1 E 21  1  ET 2   1    2  E1 E 21  2  ET 2 T E1  T E1  (36) Matrices E1 and E 2 posses the symmetries T ET E1  E2E1  I 2 (37) Therefore, the layer... (Ursin&Stovas, 2002) 13 Acoustic Waves in Layered Media - From Theory to Seismic Applications The symmetry equations (37) imply that CDT  I (56) Using equation (55), the reflection and transmission matrices are given by TD   c 11  c22 det C c 21  c12 det C  2   det(C  D)det C  c12  c 21 det C c22  c 11 det C  2 2 2 2   det C   1  c 11  c12  c 21  c22 1  RD  det(C  D)det C  2  c11c 21. .. displacement velocity - stress vector, the superscript "T " indicating the transpose, and the matrix M has the blocked structure,  0 A M   , B 0  (30) composed of the 2x2 symmetric matrices A and B are given by   c3 31 A 1   pc13c33   p  , B   , 1     p  c 11  c c    p c55   pc13c3 31 2 2 1 13 33 ( 31) 10 Waves in Fluids and Solids which are dependent on the horizontal slowness... where 1 TD  Q 22  Q 21Q 11 Q12 , (20) 1 R U  Q 21Q 11 , ( 21) 1 R D   Q 11 Q12 , (22) 1 TU  Q 11 (23) The matrices R U , TU and, R D , TD are called the reflection and transmission matrices for upand down-going wave, respectively (Figure 1) From equations (20)-(23), the amplitude propagator matrix can be rewritten as  T Q R T 1 1  TU R D U U 1 1 TD  R U TU R D U    (24) The matrix . of waves coming to the stack of layers: 1N  u and 0 d , 10 01 NDU DU N          dTRd uRTu , (19 ) where 1 22 21 11 12D  TQ QQQ, (20) 1 21 11U  RQQ, ( 21) . sub-vectors, corresponding to up- and down-going waves of different type, 0 0 0     u w d , 1 1 1 N N N        u w d . (17 ) Therefore, we can write 10 11 21 10 21 22 N N    . equations (4) and (5) according to the following relations    11 2 2 12 12 12 3 1 ,, ,, 8 ipxpxt f pp fxxte dxdxdt           ,     11 2 2 2 12 1 2 1 2 ,, ,, ipxpxt f xxt