Cover Page



Title Page


About the Author


Chapter 1: Introduction

Chapter 2: Physics of Acoustics and Acoustical Imaging

2.1 Introduction

2.2 Sound Propagation in Solids

2.3 Use of Gauge Potential Theory to Solve Acoustic Wave Equations

2.4 Propagation of Finite Wave Amplitude Sound Wave in Solids

2.5 Nonlinear Effects Due to Energy Absorption

2.6 Gauge Theory Formulation of Sound Propagation in Solids

Chapter 3: Signal Processing

3.1 Mathematical Tools in Signal Processing and Image Processing

3.2 Image Enhancement

3.3 Image Sampling and Quantization

3.4 Stochastic Modelling of Images

3.5 Beamforming

3.6 Finite-Element Method

3.7 Boundary Element Method

Chapter 4: Common Methodologies of Acoustical Imaging

4.1 Introduction

4.2 Tomography

4.3 Holography

4.4 Pulse–Echo and Transmission Modes

4.5 Acoustic Microscopy

Chapter 5: Time-Reversal Acoustics and Superresolution

5.1 Introduction

5.2 Theory of Time-Reversal Acoustics

5.3 Application of TR to Medical Ultrasound Imaging

5.4 Application of Time-Reversal Acoustics to Ultrasonic Nondestructive Testing

5.5 Application of TRA to Landmine or Buried Object Detection

5.6 Application of Time-Reversal Acoustics to Underwater Acoustics

Chapter 6: Nonlinear Acoustical Imaging

6.1 Application of Chaos Theory to Acoustical Imaging

6.2 Nonclassical Nonlinear Acoustical Imaging

6.3 Modulation Method of Nonlinear Acoustical Imaging

6.4 Harmonic Imaging

Chapter 7: High-Frequencies Acoustical Imaging

7.1 Introduction

7.2 Transducers

7.3 Electronic Circuitry

7.4 Software

7.5 Applications of High-Frequencies In Vivo Ultrasound Imaging System

7.6 System of 150 MHz Ultrasound Imaging of the Skin and the Eye

7.7 Signal Processing for the 150 MHz System

7.8 Electronic Circuits of Acoustical Microscope

Chapter 8: Statistical Treatment of Acoustical Imaging

8.1 Introduction

8.2 Scattering by Inhomogeneities

8.3 Study of the Statistical Properties of the Wavefield

8.4 Continuum Medium Approach of Statistical Treatment

Chapter 9: Nondestructive Testing

9.1 Defects Characterization

9.2 Automated Ultrasonic Testing

9.3 Guided Waves Used in Acoustical Imaging for NDT

9.4 Ultrasonic Technologies for Stress Measurement and Material Studies

9.5 Dry Contact or Noncontact Transducers

9.6 Phased Array Transducers

Chapter 10: Medical Ultrasound Imaging

10.1 Introduction

10.2 Physical Principles of Sound Propagation

10.3 Imaging Modes

10.4 B-scan Instrumentation

10.5 C-scan Instrumentation

10.6 Tissue Harmonic Imaging

10.7 Elasticity Imaging

10.8 Colour Doppler Imaging

10.9 Contrast-Enhanced Ultrasound

10.10 3D Ultrasound Medical Imaging

10.11 Development Trends

Chapter 11: Underwater Acoustical Imaging

11.1 Introduction

11.2 Principles of Underwater Acoustical Imaging Systems

11.3 Principles of Some Underwater Acoustical Imaging Systems

11.4 Characteristics of Underwater Acoustical Imaging Systems

11.5 Imaging Modalities

11.6 A Few Representative Underwater Acoustical Imaging System

11.7 Application of Robotics to Underwater Acoustical Imaging

Chapter 12: Geophysical Exploration

12.1 Introduction

12.2 Applications of Acoustical Holography to Seismic Imaging

12.3 Examples of Field Experiments

12.4 Laboratory Modelling

12.5 Techniques of Image Processing and Enhancement

12.6 Computer Reconstruction

12.7 Other Applications of Seismic Holography

12.8 Signal Processing in Seismic Holography

12.9 Application of Diffraction Tomography to Seismic Imaging

12.10 Conclusions

Chapter 13: Quantum Acoustical Imaging

13.1 Introduction

13.2 Optical Piezoelectric Transducers for Generation of Nanoacoustic Waves

13.3 Optical Detection of Nanoacoustic Waves

13.4 Nanoimaging/Quantum Acoustical Imaging

13.5 Generation and Amplification of Terahertz Acoustic Waves

13.6 Theory of Electron Inversion and Phonon Amplification Produced in the Active SL by Optical Pumping

13.7 Source for Quantum Acoustical Imaging

13.8 Phonons Entanglement for Quantum Acoustical Imaging

13.9 Applications of Quantum Acoustical Imaging

Chapter 14: Negative Refraction, Acoustical Metamaterials and Acoustical Cloaking

14.1 Introduction

14.2 Limitation of Veselago’s Theory

14.3 Multiple Scattering Approach to Perfect Acoustic Lens

14.4 Acoustical Cloaking

14.5 Acoustic Metamaterial with Simultaneous Negative Mass Density and Negative Bulk Modulus

14.6 Acoustical Cloaking Based on Nonlinear Coordinate Transformations

14.7 Acoustical Cloaking of Underwater Objects

14.8 Extension of Double Negativity to Nonlinear Acoustics

Chapter 15: New Acoustics Based on Metamaterials

15.1 Introduction

15.2 New Acoustics and Acoustical Imaging

15.3 Background of Phononic Crystals

15.4 Theory of Phononic Crystals – The Multiple Scattering Theory (MST)

15.5 Negative Refraction Derived from Gauge Invariance (Coordinates Transformation) – An Alternative Theory of Negative Refraction

15.6 Reflection and Transmission of Sound Wave at Interface of Two Media with Different Parities

15.7 Theory of Diffraction by Negative Inclusion

15.8 Extension to Theory of Diffraction by Inclusion of General Form of Mass Density and Bulk Modulus Manipulated by Predetermined Direction of Sound Propagation

15.9 A New Approach to Diffraction Theory – A Rigorous Theory Based on Material Parameters

15.10 Negative Refraction Derived from Reflection Invariance (Right-Left Symmetry) – A New Approach to Negative Refraction

15.11 A Unified Theory for Isotropy Invariance, Time Reversal Invariance and Reflection Invariance

15.12 Application of New Acoustics to Acoustic Waveguide

15.13 New Elasticity

15.14 Nonlinear Acoustics Based on Metamaterial

15.15 Ultrasonic Attenuation in Acoustic Metamaterial

15.16 Applications of Phononic Crystal Devices

15.17 Comparison of the Significance of Role Played by Gauge Theory and MST in Metamaterial – A Sum-up of the Theories of Metamaterial

15.18 Impact of New Acoustics Compared with Nonlinear Acoustics

15.19 Conclusions

Chapter 16: Future Directions and Future Technologies


Title Page

About the Author

Woon Siong Gan obtained his BSc in physics in 1965, DIC in acoustics and vibration science in May 1967 and PhD in acoustics in February 1969, all from the Physics Department of Imperial College, London. He did his postdoctoral works in Imperial College London, Chelsea College London and the International Centre for Theoretical Physics, Trieste, Italy. He is a senior life member of the IEEE, a fellow of the Institute of Engineering and Technology, UK, a fellow of the Institute of Acoustics, UK, a fellow of the Southern African Acoustics Institute, a fellow of the Institution of Engineers, Singapore, a senior member of the American Institute of Ultrasound in Medicine and a member of the Acoustical Society of America since 1969.

He is a founding president of the Society of Acoustics (Singapore), founded in 1989, and a former director of the International Institute of Acoustics and Vibration (IIAV).

He is also a founder of the Signal Processing Singapore Chapter of the Institute of Electrical and Electronics Engineers, USA. He was an associate professor in the physics department of Nanyang University, Singapore, from 1970 to 1979. He was a practicing acoustical consultant from 1979 to 1989. He founded Acoustical Technologies Singapore Pte Ltd in 1989. It is an R&T company in ultrasonics technologies and has developed and patented the scanning acoustic microscope (SAM) and the surface acoustic wave (SAW) devices. So far, it is the only ultrasonic technologies company in Singapore, specializing in ultrasound imaging.

He is a theorist and has published several papers and book chapters on acoustical imaging, active noise cancellation and the application of gauge invariance to acoustics.

He published the paper Gauge Invariance of Acoustic Fields in 2007. This has been experimentally verified by the fabrication of acoustical metamaterials, which shows the invariance of the acoustic field equation to the simultaneously negative mass density and negative bulk modulus. This has been applied to negative refraction with the fabrication of perfect acoustical lens. He also made the important discovery that negative refraction is a special case of the coordinate transformations (gauge invariance) usually applied to cloaking, when the parity or the determinant of the direction cosines or transformation matrix equals –1. His current work is the development of the new field ‘New Acoustics’ based on acoustical metamaterials. This amounts to rewriting the solutions of the acoustic wave equations when the positive mass density and positive compressibility are replaced by negative mass density and negative compressibility and the solutions of the acoustic wave equations based on the bandgap properties of phononic crystals. This will involve refraction, diffraction and scattering, the three basic mechanisms of sound propagation in solids enabling the control and manipulation of direction of sound propagation in solids and give rise to several new phenomena and applications in the form of novel acoustical devices, and hence, the term ‘New Acoustics” coined by the author.


Acoustic waves offer very different possibilities for imaging from light. Like X-rays, they can penetrate opaque media. This is why they are used for medical applications and for nondestructive testing. A wide frequency range is available. The human body can sense only one octave in the visible spectrum, but about eight octaves of sound, and ultrasound extends this range to much higher frequencies. Acoustic waves travel typically five orders of magnitude slower than electromagnetic wave, which means that submicron wavelengths can be achieved with frequencies of a few gigahertz. Transducers can be used to convert electrical signals to acoustic waves and vice versa, and the signals can be generated and processed using a full range of digital techniques.

The propagation of acoustic waves can be rich and subtle. In fluids, acoustic waves are longitudinal over all but the smallest distances. Solids can also support shear waves, with two orthogonal polarizations. In the proximity of a surface, combined shear and longitudinal waves can propagate, and they can couple into a fluid in contact with the surface. Solids that are anisotropic, such as crystals or composites, can exhibit rich phenomena, such as beam steering where the direction of propagation is not perpendicular to the wavefronts. The acoustic properties of different media can vary hugely, and this can lead to strong scattering, for example from fine cracks in solids. As well as imaging the geometry of objects, their mechanical properties can be probed. Many materials exhibit nonlinear properties at experimentally accessible acoustic amplitudes.

Exciting applications are opened up by combining a deep understanding of the propagation of acoustic waves with sophisticated instrumentation for generating and detecting them. For example, an atomic force microscope can be used to probe acoustic fields with nanometre resolution by exploiting the nonlinear interaction between the tip and a surface. Scanned lenses and arrays can be used to form diffraction-limited images with resolution from a micrometre in an acoustic microscope to a millimetre or so in medical imaging and nondestructive testing, and greater scales still in sonar and geophysics.

All of this calls for a deep understanding of the propagation of acoustic waves in fluids and in solids, and a full appreciation of the instrumentation that has been developed. Dr Gan has written a book that aims to address this need comprehensively. My hope is that it will lead to better-informed and more widespread use of the rich resources of acoustic imaging.

G.A.D. Briggs
14 May 2011



Acoustical imaging is a multidisciplinary subject covering physics, mechanical engineering, electrical engineering, biology and chemistry. Imaging carries information; it is the procedure for recording information; and there is a saying that an image is worth a thousand equations. There are several forms of imaging modalities using various means of carrying information such as light waves, X-rays, γ-rays, electron beams, microwaves and sound waves. Of these, sound waves, like X-rays, have the capability of penetrating an opaque medium thus enabling the interior of structures to be imaged, due to the propagation of vibration to the interior of the material, and sound waves are generated by vibration. Compared with other imaging modalities, sound waves are safe with no radiation hazards.

This book is the only textbook and reference book that covers all engineering applications under one roof. It is also unique as it presents the latest developments and forefront research in acoustical imaging in the areas of elasticity imaging, time reversal acoustics applied to acoustical imaging, nonlinear acoustical imaging, stochastic and statistical treatments of multiple scattering effects in acoustical imaging, the application of negative refraction to acoustical imaging, and the new field of ‘new acoustics’ founded by the author. The book will therefore be of great interests to practising engineers and researchers. To reflect the engineering nature of the book, there are chapters on such topics as: signal processing and image processing, nondestructive evaluation, underwater acoustics and geophysical exploration.

Acoustical imaging is an old discipline. Various animals have the capability of acoustical imaging. Echo-locating bats, for example, can catch their prey in complete darkness. They utter twittering sounds, too high-pitched for human ears to detect, and process the echoes of this sound from nearby objects to avoid colliding with obstructions. This gives the bats an acoustical image of its surroundings. With a specialized larynx, unusually sensitive ears and a highly developed audio cortex, bats can quickly and safely navigate through the various potential obstructions in the darkness of a cave. Using the same principle of acoustical imaging, dolphins and whales can navigate the murky waters of the ocean. The acoustical imaging ability of animals is the basis of the principles of sonar – the use of pulse–echo technology for underwater viewing in the ocean. The significance and motivation for the development of sonar was: (1) the sinking of the Titanic, the world’s largest ship, by colliding with an iceberg when moving at high speed about 1600 miles northeast of New York City on 15 April 1912; and (2) the German U-boat threat to French shipping in World War I.

Lord Rayleigh and O.P. Richardson had thought of using ultrasonic waves for underwater imaging, and the tragic shipwreck of the Titanic stimulated many new activities on how to prevent accidents of this kind in the future. In 1912, Hiram S. Maxim, an American engineer and inventor, was inspired by the techniques employed by bats. He proposed that ships could be protected from collisions with icebergs and other ships by generating underwater sound pulses and detecting their echoes. Shortly afterwards, L.F. Richardson filed a patent in 1912 for a device that would produce sound waves in either air or water and detect echoes from distant objects. R.A. Fessenden filed a patent in the United States early in 1913 for a similar invention. One year later an iceberg was successfully detected at a distance of 2 km using Fessenden’s invention.

Further momentum for the development of more advanced and sophisticated underwater detection equipment arose during World War I due to the immense destructive power of German submarines. Paul Langevin, an outstanding French physicist, was commissioned by his government to find an effective method of detecting the submarines. M.C. Chilowski, an engineer, had developed ultrasonic equipment for the French navy, but its acoustic intensity was much too weak to be effective. Heading a joint US, British and French venture, Langevin looked into the problem of how to increase the acoustic intensity in the water. Within three years, he succeeded in generating a higher acoustic intensity by means of piezoelectric transducers operating at resonance. By 1918, active systems for generating, receiving, and analyzing returned acoustic echoes were developed and proved useful in antisubmarine activities.

The above acoustical imaging systems are intended to image structures within the vast domains of the ocean. Extensive research programmes were needed before other acoustic systems could be applied to the imaging of small-scale systems, such as the tiny interior structures within regions of interest in industry, in hospitals and in laboratories. One of the most important of these programmes was that of the Russian scientist Sokolov [1] whose works started in the 1920s. He was one of the first to recognize and systematically explore the use of ultrasound to image the internal structures of optically opaque objects. Some of his systems were designed to image inhomogeneities, such as cracks, flaws and voids within manufacturing parts. In one of his systems, the inhomogeneities were made visible by reflecting collimated light from a liquid surface in a manner similar to that of liquid-surface acoustical holography [2]. The system provided a means of encoding image information to enable the image to be read out in real time by light diffracted from the sound. The method was a precursor of acoustical holography, and predated Denis Gabor’s invention of holography.

In Langevin’s system, the ultrasonic transmitter emitted a pulse, and the amplitude of the echo, or reflected pulse, was used to produce the acoustical image. In Sokolov’s system, the ultrasonic transmitter emitted continuous waves and the amplitude and phase of the transmitted waves were both used to produce the image. Since Langevin and Sokolov several acoustical imaging systems have been invented with various features and certain degrees of success.

Presently, the acoustical imaging systems can be classified into three main types: pulse–echo, phase–amplitude and amplitude mapping. Examples of pulse–echo systems are B-scan and C-scan systems for medical imaging and nondestructive evaluation, linear array systems for geophysical exploration and seismology, and sonar systems for underwater acoustics. Acoustical holography is an example of a phase–amplitude system. A typical example of an amplitude-mapping system is acoustical microscopy [3] for nondestructive evaluation, failure analysis, material studies, and biomedical imaging and analysis.

Pulse–echo techniques [4] will involve knowhow in transducer technology, both single element and array types for the generation of ultrasound, electronics such as pulser–receiver for the transmission and receiving of ultrasonic signals, data acquisition card for the capturing and digitization of analogue ultrasonic signals for computation, and software for interfacing the hardware parts. The advance in electronics and digital-processing techniques over the last few decades has given rise to major improvements in the systems available and permitted the development of new and significant scanning and processing methods. This book reviews the latest technology improvements and describes new concepts and approaches that have emerged from the research laboratories where work is being done in this segment.

An example of phase–amplitude acoustical imaging is the acoustical holographic system [2], which originated from Denis Gabor’s Nobel prize works in holography in 1948 [5]. Gabor’s original purpose was in the improvement of the resolution of the electron microscope to 1 Å to view the atom. He invented holography as a lensless two-step imaging process in which a hologram would be generated by a scattered electron beam and reconstruction would take place by means of an optical beam. Holographic techniques, however, are not limited to electron or optical beams. Coherence beams are the critical requirement. Holographic systems record both the amplitude and the phase of the scattered beam, and the phase information gives rise to three-dimensional images.

Acoustical microscopic [3] systems were invented in 1974. Unlike the previous two systems, in which the frequencies were typically well below 10 MHz, the acoustic microscopes use much higher frequencies, ranging from tens of megahertz to gigahertz.

As a unique feature, this book includes all the latest inventions in acoustical imaging systems and the forefront research in acoustical imaging systems after the three main types of imaging systems described previously. These will include elasticity imaging, nonlinear acoustical imaging for nondestructive evaluation, time reversal acoustics in acoustical imaging, stochastic and statistical treatment of acoustical imaging, the application of chaos theory to acoustical imaging, and the application of negative refraction to acoustical imaging.

Acoustical imaging is the study of sound propagation in solids, making use of the mechanical and elastic properties to image the interior structure of solids. It will build on the theory of elasticity, the theory of diffraction, the theory of single and multiple scattering, time reversal acoustics, and gauge invariance approach to acoustic fields. Recently, I have pioneered the application of gauge theory and symmetries as a framework to describe sound propagation in solids, and an introduction to this new subject will be given in Chapter 15 of this book.

Signal processing and image processing are important topics of engineering interest. They can be applied to all three of the main engineering applications of acoustical imaging: (1) nondestructive evaluation, (2) underwater acoustics, and (3) geophysical exploration. Examples of some techniques of signal processing are spatial deconvolution, histogram-based amplitude mapping, operator construction, quantization errors and wavefield orthogonalization. Image-processing techniques will help in image understanding, which covers texture analysis and tissue characterization. Some topics of image processing are image sampling and quantization, image transforms, image representations by stochastic models, image enhancement, image filtering and restoration, and image reconstruction from projections.

This book is particularly intended for practising engineers and researchers. The reader will study all the key areas of the engineering applications of acoustical imaging under one cover. Looking forward to the prospects of acoustical imaging, its share of the global market is already catching up with X-rays. It also has the advantages of no radiation hazards and is safe to use regularly with long exposure over a period of time. It also has the capability of viewing minute cracks in the texture of materials, which is outside the ability of an X-ray. With the explosive advancement and application of nanotechnology, acoustical imaging will play an even more important role in the forthcoming decades.


1. Sokolov, S. USSR Patent no. 49 (31 August 1936); British Patent no. 477 139, 1937; US Patent no. 21 64 125, 1939.

2. Mueller, R.K. and Sheridon, N.K. (1966) Sound holograms and optical reconstruction. Appl. Phys. Lett., 9, 328.

3. Korpel, A. (1974) Acoustic Microscopy in Ultrasonic Imaging and Holography (eds G.W. Stroke et al.), Plenum Press, New York.

4. Wells, P.N.T. (1977) Biomedical Ultrasonics, Academic Press, New York.

5. Gabor, D. (1948) A new microscopic principle. Nature, 161, 777.


Physics of Acoustics and Acoustical Imaging

2.1 Introduction

Acoustical imaging involves the study of sound propagation in solids or fluid models. There are various formulations. On the one hand, some studies are based on diffraction theory [1]; and, on the other, some works are based on the acoustical equations of motion and the theory of elasticity [2]. However, these methods are all limited to treatments based on linear acoustic waves. Our presentation will also include large amplitude sound propagation and an introduction to the formulation of gauge theory, which involves symmetries, Galilean transformations, and covariant derivatives.

2.2 Sound Propagation in Solids

2.2.1 Derivation of Linear Wave Equation of Motion and its Solutions

Our work emphasizes the mechanical and elastic properties of sound waves. We start with the propagation of linear sound waves or infinitesimal amplitude sound waves in solids. First, the acoustic field equations of motion are derived. There are two basic field equations: the first is obtained from Newton’s laws of motion in mechanics, and the second from Hooke’s law in the theory of elasticity. The first field equation expresses Newton’s law of motion, written as

(2.1) Numbered Display Equation

The second field equation is the strain–displacement relation, related to Hooke’s law, and is written as

(2.2) Numbered Display Equation

where T is the stress, u is the displacement, F is the body force, S is the strain and ρ is the density of the medium. In order to solve for the variables u and T, a second equation is necessary. This is given by Hooke’s law in the theory of elasticity, which states that strain is linearly proportional to stress. Thus

(2.3) Numbered Display Equation

where i, j, k, l = x, y, z, with an implicit summation convention over the repeated subscripts k and l. The microscopic spring constants cijkl in equation (2.3) are called the elastic stiffness constants.

We consider a source-free region, so that F = 0. The next step is to eliminate T from equations (2.1) and (2.3). From equations (2.2) and (2.3) together, T = cijkl , if it is only in one dimension, the x direction is chosen. Substituting in equation (2.1), we obtain

(2.4) Numbered Display Equation

which is known as the Christoffel equation.

Equation (2.4) denotes a travelling wave, and its solution is

(2.5) Numbered Display Equation

which gives

(2.6) Numbered Display Equation

The phase velocity is given by v = ω/k. Thus, for transverse (or shear) waves, the velocity is

(2.7) Numbered Display Equation

2.2.2 Symmetries in Linear Acoustic Wave Equations and the New Stress Field Equation

Equation (2.2) can also be written in terms of the particle velocity and compliance as

(2.8) Numbered Display Equation

where s is compliance.

Acoustic wave equations can be obtained by eliminating either T or v from the acoustic field equations. Usually the stress field is eliminated since it is a tensor quantity and consists of six field components rather than the three associated with a vector field.

For infinitesimal amplitude sound waves, the lossless acoustic field equations are given by equations (2.1) and (2.2). We shall now eliminate the velocity field from equations (2.1) and (2.8).

Differentiating equation (2.8) with respect to t

(2.9) Numbered Display Equation

with F = 0 for a source-free region, and taking the divergence of both sides of (2.1)

(2.10) Numbered Display Equation

By the insertion of (2.9) we also have

Unnumbered Display Equation


(2.11) Numbered Display Equation

This is a new stress equation. The potential and the applications of this equation have yet to be explored.

We thus discover an important property: the acoustic wave equations (2.4) and (2.11) are symmetrical in u and T. This symmetry gives rise to several simplifications in solving acoustic wave equations.

2.3 Use of Gauge Potential Theory to Solve Acoustic Wave Equations

By analogy with the electromagnetic wave field, we can also represent the acoustic particle velocity field in terms of the gauge potentials of gauge theory – that is, in terms of the scalar potential φ and the vector potential A. For isotropic media, which are always nonpiezoelectric, the Christoffel equation can be written as

(2.12) Numbered Display Equation

for an isotropic medium. There is a theorem which states that for isotropic solids, there are only two elastic constants C11 and C44. This is a consequence of the fact that for an isotropic solid, the elastic properties are symmetrical in all directions. This yields plane wave solutions with harmonic time variation. To obtain the general equation for plane wave solutions, the substitutions

Unnumbered Display Equation

are inserted. This gives

(2.13) Numbered Display Equation


(2.14) Numbered Display Equation

where the vector identity

(2.15) Numbered Display Equation

has been used to rearrange the terms. Solutions of equation (2.14) are obtained by using a gauge theory formulation expressing v in terms of the gauge potentials: the scalar potential φ and the vector potential A.

(2.16) Numbered Display Equation

Substitution of equation (2.16) in equation (2.14) gives

(2.17) Numbered Display Equation

since For the second term, the quantity in brackets is set equal to the gradient of an arbitrary function f

(2.18) Numbered Display Equation

The application of identity (2.15) will convert equation (2.18) into

(2.19) Numbered Display Equation

where . Since f is arbitrary, it can always be chosen to cancel ⋅ A in the first term on the left. The vector potential can thus be chosen as a solution to the vector potential wave equation

(2.20) Numbered Display Equation

The first term in equation (2.17) is made zero simply by requiring that the scalar potential satisfy the following scalar potential wave equation

(2.21) Numbered Display Equation

Equations (2.20) and (2.21) show that the linear wave equations are symmetrical in and A, as in the case for electromagnetic waves. equations (2.20) and (2.21) are of the same form as the Helmholtz wave equation, which confirms the analogy.

2.4 Propagation of Finite Wave Amplitude Sound Wave in Solids

Acoustical imaging concerns the propagation of sound waves in solids. In real-life situations, and in practical circumstances, the sound waves are usually of finite amplitude. In the previous sections our acoustic equations of motion or acoustic field equations were for infinitesimal wave amplitudes. In the present section, we extend our treatment to finite wave amplitudes. The equations of motion, and the subsequent sound wave equations, will be nonlinear in nature. There are two general sources of nonlinearity: one is known as the kinematic or convective nonlinearity, which is independent of the material properties; and the other is the inherent physical or geometric nonlinearity of the solid. Here, we will deal with the derivation of the finite amplitude or nonlinear acoustic equations of motion and their solutions. The two major works in this area are those of Zarembo and Krasil’nikov [3] and Thurston and Shapiro [4] and our account will be based on these two investigations.

When dealing with finite amplitude sound waves in solids, we have to deal with the effects of nonlinearity and two mechanisms must usually be considered: (1) higher-order elasticity theory and (2) energy absorption or the attenuation of sound waves in solids.

2.4.1 Higher-Order Elasticity Theory

Since finite amplitude sound wave waves involve finite displacements, the stress in a solid is no longer linearly related to the strain, and Hooke’s law is no longer valid.

The elastic energy stored in a deformed isotropic or anisotropic solid can be written in tensor notation as

(2.22) Numbered Display Equation

where i, j, k, l, m, n = 1, 2 or 3, cijklmn are third-order elastic constants or stiffness, Sij, etc. are elastic strains. If only the first term in equation (2.22) is included, we recover linear elasticity theory and cijkl is the second-order elastic constants or stiffness as it contains elastic strain products of the second degree. In first-order elasticity theory, only two Lamé constants, λ and μ, are required for an isotropic body.

Truesdell [3] has shown that, in any isotropic elastic material, sound waves travelling down a principal axis of stress are in either purely longitudinal or purely transverse modes.

2.4.2 Nonlinear Effects

The nonlinear effects in the propagation of finite amplitude sound waves in solids may arise from the following causes: (1) large wave amplitudes, giving rise to finite strains; (2) a medium amplitude behaving locally in a nonlinear manner due to the presence of various energy-absorbing mechanisms.

Nonlinear propagation differs from linear elastic waves in that the initially sinusoidal longitudinal stress wave of a given frequency becomes distorted as propagation proceeds, and energy is transferred from the fundamental to the harmonics that develop. The degree of distortion and the strength of harmonic generation depend directly on the amplitude of the initial wave. A pure-mode longitudinal nonlinear wave may propagate as such, while a pure transverse nonlinear wave will necessarily be accompanied by a longitudinal wave during propagation. In addition, a nonlinear transverse wave, unlike the nonlinear longitudinal wave, does not distort when propagating through a defect-free solid.

Nonlinear sound waves can interact with other waves in a solid, and in the region of the interaction of two ultrasonic beams, a third ultrasonic beam may be generated.

2.4.3 Derivation of the Nonlinear Acoustic Equation of Motion

When a finite amplitude sound wave propagates in solids, large displacements are incurred and the stress is no longer linearly related to the strain. Thurston and Shapiro [4] considered the simplified case of one-dimensional motion in an isotropic solid or along certain directions in anisotropic media. They obtained a higher-order acoustic equation of motion in the form

(2.23) Numbered Display Equation

where x is the Lagrangian coordinate in the direction of motion of a particle, u is the displacement, ρ is the density of the medium, M2 = K2, M3 = K3 + 2K2, M4 are linear combinations of second-, third- and fourth-order elastic coefficients, while K2 and K3 are related to the second- and third-order elastic constants.

If the fourth and higher orders are omitted, equation (2.23) reduces to the following approximate form:

(2.24) Numbered Display Equation

Equation (2.23) is obtained by considering only one aspect of the nonlinear effects of the propagation of finite amplitude sound waves in solids, namely the elasticity effect.

Another important aspect of nonlinearity is the energy absorption or attenuation of sound waves in solids. To account for this, an additional term has to be included in equation (2.23), which modifies it to (see Stephens [5])

(2.25) Numbered Display Equation

where denotes the speed of propagation of a wave of infinitesimal amplitude.

In general, energy absorption increases with frequency, and the wave front will attain maximum steepness when the transfer of energy to higher harmonics due to nonlinearity is just equal to the increase in absorption at the higher frequencies. It is only a relative stabilization of the wave profile as, due to damping, the wave gradually returns towards its initial sinusoidal shape.

2.4.4 Solutions of the Higher-Order Acoustics Equations of Motion

The usual method of solving higher-order acoustic equations of motion is to apply perturbation theory, with a sound source excitation of u(0, t) = u0sinωt. According to Zarembo and Krasil’nikov [3], equation (2.24) will produce a second harmonic

(2.26) Numbered Display Equation


Unnumbered Display Equation

in which A , B and C are third-order elastic moduli, ρ0 is a constant density in the unstressed configuration, cl is the propagation speed of the linearized (small amplitude) longitudinal elastic wave, is a retarded time variable, a is the original position in the unstressed state and is a materials coordinate.

As the amplitude of the second harmonic contains β and the third-order elasticity (TOE) modulus, it can therefore be used to measure the TOE. The perturbation method leading to the solution in equation (2.26) was followed using a truncation of the exact constitutive equation and neglecting the generation of second harmonics from coefficients of the fourth and higher orders.

With the energy absorption included, as given by equation (2.25), the solution is given by Zarembo and Krasil’nikov [3] as

(2.27) Numbered Display Equation

where is the shear viscosity coefficient, ξ is the bulk viscosity coefficient, and χ = , in which K is the thermal conductivity, T0 is the ambient temperature, αT is the thermal expansion coefficient, Cp = specific heat per unit volume at constant pressure, and κ is the bulk modulus.

An isotropic elastic solid supports transverse or shear wave motion in addition to supporting longitudinal waves.

2.5 Nonlinear Effects Due to Energy Absorption

When finite amplitude sound waves propagate in solids, there is also a nonlinear effect due to the energy absorption, which causes attenuation of the sound waves. This also results in a departure from Hooke’s law. Of the various mechanisms involved with energy absorption, we first consider thermal conductivity.

2.5.1 Energy Absorption Due to Thermal Conductivity

Energy absorption due to thermal conductivity is generally negligible except in metals at frequencies of about 103 MHz or more. The thermal motion of the atoms about their mean positions in a solid can be expressed as a superposition of large numbers of mechanical waves. These waves are known as Debye waves or phonons as they are termed in quantum mechanics. In a nonmetal, heat energy is carried entirely by these thermal phonons. The resistance to heat flow in the presence of a temperature gradient arises from the fact that a phonon wave loses its momentum, or is attenuated, owing to its interaction with the phonons. This interaction between the applied sound waves or phonons and the thermal phonons produces measurable attenuation of the former at frequencies above 103 MHz (Bhatia [6]).

2.5.2 Energy Absorption Due to Dislocation

At temperatures above 20°K for metal crystals, and at all temperatures for nonmetallic crystals, most of the energy absorption at ordinary ultrasonic frequencies is believed to arise from the interaction of sound waves with dislocations – a type of extended fault in the crystal. In a polycrystalline material, the absorption is greater than that in a single crystal of the same substance. Usually, the greater the additional absorption, the greater the elastic anisotropy. The main causes are (a) thermoelastic damping due to the flow of heat across the grain boundaries and, particularly, (b) the scattering of sound waves by individual grains, which is important in the megacycle frequency range because the crystal axes of different grains are differently oriented with respect to a fixed set of axes in space and, hence, possess different elastic constants for wave propagation in a given direction. Also, reflection and scattering occur at the grain boundaries.

2.6 Gauge Theory Formulation of Sound Propagation in Solids

The acoustic equations of motion were derived for the case of a stationary medium, but in real-world situations, the medium is usually moving. This applies in particular to the case where the sound wave is propagating in a solid, and the unstressed state of the material is evolving with time. Galilean transformation, or Galilean symmetry, is the type of gauge transformation applicable to the propagation of sound in solids. For sound propagation in solids, the Galilean transformation should additionally include both translational and rotational symmetry. Kambe [7] derived a gauge theory formulation for ideal fluid flows based on Galilean transformations and covariant derivatives, which are properties of the gauge transformation and are intrinsic to the acoustic equation of motion. Here, we extend the gauge principles to sound propagation in solids. Covariant derivatives and Galilean transformations are gauge transformations. The analogy in the electromagnetic counterpart is that the covariant derivative is also intrinsic to Maxwell’s equations. However, due to the different nature of sound waves and electromagnetic waves, the covariant derivative for Maxwell’s equations leads to the Lorentz transformation, and the covariant derivative for the acoustic equation of motion leads to the Galilean transformation. Of course, the Lorentz transformation reduces to the Galilean transformation when the velocity of the medium is much less than the velocity of light.

First we shall give a brief description of the gauge principle. In gauge theory, there is a global gauge invariance and a local gauge invariance. Local gauge invariance is more stringent than global gauge invariance. Weyl’s gauge principle states that when the original Lagrangian is not locally gauge invariant, a new gauge field must be introduced in order to satisfy local gauge invariance, and the Lagrangian is then to be altered by replacing the partial derivative with the covariant derivative. The introduction of a covariant derivative is necessary for local gauge invariance, as well as to satisfy the Galilean transformation. This can be represented as

(2.28) Numbered Display Equation

where Dt is the covariant derivative, and G is the new gauge field.

We will use the Galilean transformation that describes sound propagation in solids. The symmetries to be investigated here are both translational and rotational. First, we consider translational symmetry without local rotation. A translational transformation from one coordinate system A to another A′ moving with a relative velocity R is called a Galilean transformation in Newtonian mechanics. The transformation law (see Figure 2.1) is defined by

(2.29) Numbered Display Equation

which is a sequence of global translational transformations with parameter t.

For local Galilean transformation, Kambe [7] has derived a covariant derivative, given as

(2.30) Numbered Display Equation

2.6.1 Introduction of a Covariant Derivative in the Infinitesimal Amplitude Sound Wave Equation

Replacing the partial derivative in equation (2.1) by the covariant derivative given by equation (2.30), we have

(2.31) Numbered Display Equation

If only one direction (say, the x direction) is chosen, with F = 0 for a source-free region, equation (2.31) reduces to a simpler form

(2.32) Numbered Display Equation

where x′ denotes the moving coordinate, given by x′ = xRt.

We realize that, with the introduction of the covariant derivative, there is an additional second term on the right-hand side of the equation. So far, no one has attempted to find an exact analytical solution for this equation.

Figure 2.1 Coordinate system moving with velocity R translationally


2.6.2 Introduction of Covariant Derivative to the Large Amplitude Sound Wave Equation

When we apply the covariant derivative (2.30) to the nonlinear wave equation given by equation (2.24), we obtain

(2.33) Numbered Display Equation

The introduction of the covariant derivative only introduces the same additional term on the right-hand side of the equation, as in the case for the linear wave equation in equation (2.32). Again, no one has yet obtained an exact analytical solution for this equation.


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