акустика / gan_ws_acoustical_imaging_techniques_and_applications_for_en

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.

2.3Use 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

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

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.

since · × A = 0 and × ϕ = 0. For the second term, the quantity in brackets is set equal to the gradient of an arbitrary function f

where vs = cρ44 . 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

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

Equations (2.20) and (2.21) show that the linear wave equations are symmetrical in ϕ 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.

2.4Propagation 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.

2.4.1Higher-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

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 Lame´ 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.

2.4.2Nonlinear 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.

2.4.3Derivation 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

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-, thirdand fourth-order elastic coefficients, while K2 and K3 are related to the secondand third-order elastic constants.

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

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])

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.

2.4.4Solutions 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 ) = u0 sin ωt. According to Zarembo and Krasil’nikov [3], equation (2.24) will produce a second harmonic

2

u(a, τ ) = u0 sin ωτ + β ωu0 a cos 2ωτ + · · · (2.26)

4cl

where

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, τ = t − a is a retarded time variable, a is the original position in the unstressed

cl

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

where α = ω2 ( 43 η + ξ + χ )/2ρ0c3l , η is the shear viscosity coefficient, ξ is the bulk viscosity coefficient, and χ = KT0 (αT κ /clCp )2, 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.

2.5Nonlinear 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.

2.5.1Energy 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]).

2.5.2Energy 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.

2.6Gauge 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

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

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

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

References

[1]Kak, A.C. and Slaney, M. (1987) Principles of Computerized Tomographic Imaging, Wiley-IEEE Press, New York, pp. 204–218.

[2]Auld, B.A. (1990) Acoustic Fields and Waves in Solids, vol. 1. Robert E Krieger Publishing Company, pp. 106–107, Florida, USA.

[3]Zarembo, L.K. and Krasil’nikov, V.A. (1971) Nonlinear phenomena in the propagation of elastic waves. Sov. Phys. Usp., 13, 778–797.

[4]Thurston, R.N. and Shapiro, M.J. (1967) Interpretation of Ultrasonic Experiments of Finite-Amplitude Waves.

Journ. Acoust. Soc. Amer., 41, 1112.

[5]Stephens, R.W.B. (1976) Finite-amplitude propagation in solids. Proceedings of the International School of Physics (Enrico Fermi), Course LXIII New Directions in Physical Acoustics, pp. 409–436.

[6]Bhatia, A.B. (1985) Ultrasonic Absorption, Dover, New York, pp. 290–370.

[7]Kambe, T. (2007) Variational Formulation of Ideal Fluid Flows According to Gauge Principle. Preprint accepted by Fluid Dynamics Research, Elsevier Science, the Netherlands.

3.1.2Some Properties of Matrices

Transpose:

If a matrix is represented as A = {a(m, n)}, where m is the row index and n is the column index, then its transpose is given as AT = {a(n, m)} meaning that the rows and columns are interchanged.

Complex Conjugate:

The complex conjugate of the matrix A is given by A = {a (m, n)}

Conjugate Transpose:

The conjugate transpose is given by

A = a (m, n)

Complex Transpose:

The complex transpose is given by

A T = a (m, n)

Identity Matrix:

The identity matrix is given by

I = {δ(m − n)}

It is a square matrix with unity along its diagonal.

Null Matrix:

0 = {0}, meaning that all elements are zero.

Matrix Addition:

A + B = {a (m, n) + b(m, n)}

where A and B have the same dimensions.

Scalar Multiplication:

αA = {αa(m, n)}

where α is a scalar quantity.

Matrix Multiplication:

k

c (m, n) = a (m, k) b(k, n)

k=1

where C = AB, A = M × K, B = K × N, C = M × N and AB = BA.

Commuting Matrices:

When AB = BA.

Determinant:

|A|, for square matrices only.