Waves in fluids and solids Part 1 pot

Dvoesherstov Atomic Vibrations and Propagation of Acoustic Waves in Heterogeneous Systems 103 Alexander Feher, Eugen Syrkin, Sergey Feodosyev, Igor Gospodarev, Elena Manzhelii, Alexand

WAVES IN FLUIDS

AND SOLIDS Edited by Rubén Picó Vila

Waves in Fluids and Solids

Edited by Rubén Picó Vila

Published by InTech

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Contents

Preface IX Part 1 Elastic Waves in Solids 1

- From Theory to Seismic Applications 3

Alexey Stovas and Yury Roganov

I Djeran-Maigre and S V Kuznetsov

V I Cherednick and M Y Dvoesherstov

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

in Granular Media: Theory and Experiments 127

Leonardo Trujillo, Franklin Peniche and Xiaoping Jia

Hefeng Dong and Jens M Hovem

N F Bunkin and V S Gorelik

Part 2 Acoustic Waves in Fluids 209

with Low Numerical Dispersion for Simulating 3D Wave Propagation 211

Dinghui Yang, Xiao Ma, Shan Chen and Meixia Wang

VI Contents

an Acoustic Wave and Levitated Microparticles 241

Ovidiu S Stoican

Bin Liang, Ying Yuan, Xin-ye Zou and Jian-chun Cheng

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In practice, the wavelength could be comparable with the layer thickness, and application of both frequency limits is no longer valid The frequency-dependent effects for acoustic waves propagating through the layered media are analyzed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 differences, elements, and other related numerical techniques are valuable tools for the simulation

finite-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

inversion procedure by perturbation of the parameters characterizing the elastic medium until the synthetic records fit the observed real data It is not advisable to apply these techniques in large-scale computation, especially for a large scale 3D simulation of seismic wave propagation because of an intensive use of Central Processing Unit (CPU) time and the requirement of a large amount of direct-access memory A 3D numerical method is proposed to effectively suppress the numerical dispersion caused by the discretization of the acoustic- and elastic-wave equations through using both the local spatial difference-operator and the fourth-order Runge-Kutta (RK) method so that the numerical technique developed in this chapter has rapid computational speed and can save the memory storage

The electrode systems generating a quadrupole electric field, having both static and variable components, under certain conditions, allow to maintain the charged particles

in a well defined region of space without physical solid contact with the wall of a container This process is sometime called levitation Usually, these kinds of devices are known as quadrupole traps The possibility to manipulate the stored microparticles by using an acoustic wave is investigated by experimental means That means both controlling their position in space and performing a further selection of the stored microparticles

There exists a special class of solid media called soft media (or weakly compressible media), for which the inequality λ>>μ is satisfied (λ and μ are the Lamé coefficients) Such media with very small shear stiffness are dynamically similar to liquids to a great extent and exhibit strongly a “water-like” characteristic The class of soft media includes many common media in the fields of scientific research and practical applications (e.g soft rubbers, tissues, or biomimetic materials), and air bubbles are often introduced due to artificial or non-artificial reasons Propagation of acoustic waves in a bubbly soft medium is particularly different from that of a usual solid medium containing bubbles

The interaction of a point-source spherical acoustic wave with a bounded obstacle possesses various attractive and useful properties in direct and inverse scattering theory More precisely, concerning the direct scattering problem, the far-field interaction of a point source with an obstacle is, under certain conditions, stronger compared to that of a plane wave In inverse scattering problems the distance of the point-source from the obstacle constitutes a crucial parameter, which is encoded in the far-field pattern and is utilized appropriately for the localization and reconstruction of the obstacle's physical and geometrical characteristics

Rubén Picó Vila

Polytechnic University of Valencia

Spain

Part 1

Elastic Waves in Solids

1

Acoustic Waves in Layered Media

- From Theory to Seismic Applications

Alexey Stovas1 and Yury Roganov2

In practice, the wavelength could be comparable with the layer thickness, and application of both frequency limits is no longer valid In this chapter, we will mainly focus on the frequency-dependent effects for acoustic waves propagating through the layered media

We show that there are distinct periodically repeated patterns consisted of the pass- and stop-bands of very complicated configuration defined in frequency-slowness or frequency-group angle domain that control the reflection and transmission responses The edges between the pass- and stop-bands result in the caustics in the group domain The quasi-shear waves in a homogeneous transversely isotropic medium could also results in the high-frequency caustics, but for the layered media, all wave modes can result in frequency-dependent caustics The caustics computed for a specific frequency differ from those observed at the low- and high-frequency limits From physics point of view, the pass-bands correspond to the effective medium, while the stop-bands correspond to the resonant medium We distinguish between the effects of scattering and intrinsic attenuation in layered media The propagation of acoustic waves in a layered medium results in the energy loss due to scattering effect The intrinsic attenuation is an additional effect which plays very important role in seismic data inversion We provide the theoretical and numerical study to compare both effects for a periodically layered medium We also investigate the complex frequency roots of the reflection/transmission responses We also derive the phase velocity approximations in a layered medium As the trial model for layered medium, we widely use the periodically layered medium with the limited number of parameters The propagation of acoustic waves through a periodic layered medium is analyzed by an eigenvalue

Waves in Fluids and Solids

4

decomposition of the propagator matrix This reveals how the velocity and attenuation of the layered medium vary as function of the periodic structure, material parameters and frequency We show that there is one more parameter controlling the wave propagation apart of the wavelength to layer thickness ratio that is the acoustic contrast between the layers Multiple scattering in finely layered sediments is important in stratigraphic interpretation in seismic, matching of well log-data with seismic data and seismic modelling Two methods have been used to treat this problem in seismic applications: the O’Doherty-Anstey approximation and Backus averaging The O’Doherty-Anstey approximation describes the stratigraphic filtering effects, while the Backus averaging defines the elastic properties for an effective medium from the stack of the layers

Using numerical examples, we show that there is a transition zone between the effective medium (low-frequency limit) and the time-average medium (high-frequency limit) and that the width of this zone depends on the strength of the reflection coefficient series Assuming that a tubidite reservoir can be approximated by a stack of thin shale-sand layers we use standard AVO-attributes to estimate net-to-gross and oil saturation Necessary input is Gassmann rock physics properties for sand and shale as well as the fluid properties for hydrocarbons Required seismic input is AVO intercept and gradient The method is based upon thin layer reflectivity modeling It is shown that random variability in thickness and seismic properties of the thin sand and shale layers does not change the AVO attributes at top and base of the turbidite reservoir sequence significantly The method is tested on seismic data from offshore Brazil, and the results show reasonable agreement between estimated and observed net-to-gross and oil saturation The methodology can be further developed for estimating changes in pay thickness from time lapse seismic data We propose the method of computation seismic AVO attributes (intercept and gradient) from ultra-thin geological model based on the SBED modelling software The SBED software is based on manipulating sine-functions, creating surfaces representing incremental sedimentation Displacement of the surfaces creates a three dimensional image mimicking bedform migration, and depositional environments as diverse as tidal channels and mass flows can be accurately recreated The resulting modelled deposit volume may be populated with petrophysical information, creating intrinsic properties such as porosity and permeability (both vertical and horizontal) The Backus averaging technique is used for up-scaling within the centimetre scale (the intrinsic net-to-gross value controls the acoustic properties of the ultra-thin layers) It results in pseudo-log data including the intrinsic anisotropy parameters The synthetic seismic modelling is given by the matrix propagator method allows us to take into account all pure mode multiples, and resulting AVO attributes become frequency dependent Within this ultra-thin model we can test different fluid saturation scenarios and quantify the likelihood of possible composite analogues This modelling can also be used for inversion of real seismic data into net-to-gross and fluid saturation for ultra-thin reservoirs

There are many other issues related to wave propagation in layered media we do not discuss in this chapter For further reading we suggest several books (Aki&Richards, 1980; Brekhovskih, 1960; Kennett, 1983; Tsvankin, 1995) that cover the problems we did not touch here

2 System of differential equations

To describe the dynamic of the wave propagation in an elastic medium, it is common to use the Hook’s law that defines the linear relation between the stress tensor ij and the strains tensor e pq,