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ELASTIC WAVES

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»- O N) O 03 N) BIBLIOTHEEK TU Delft P 1822 5348 C 557126

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PROEFSCHRIFT

TER VERKRIJGING VAN DE GRAAD VAN DOCTOR

IN DE TECHNISCHE WETENSCHAPPEN AAN DE

TECHNISCHE HOGESCHOOL DELFT, OP GEZAG

VAN DE RECTOR MAGNIFICUS PROF. DR. IR.

H. VAN BEKKUM, VOOR EEN COMMISSIE

AAN-GEWEZEN DOOR HET COLLEGE VAN DEKANEN,

TE VERDEDIGEN OP WOENSDAG 26 NOVEMBER 1975

TE 16.00 UUR

door

TAN TIK HING

elektrotechnisch ingenieur

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Laboratory of Electromagnetic Research, Delft, The Netherlands, under the financial support of the Netherlands organization for the advancement of pure research (Z.W.O.).

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SUMMARY 13 Chapter I

SCATTERING OF ELASTIC WAVES BY ELASTICALLY

TRANSPARENT OBSTACLES (INTEGRAL-EQUATION METHOD)* 17

1. Introduction 17 2. Basic relations in the theory of elastic 19

waves

3. Exchange of acoustic energy 22 A. Elastodynamic reciprocity relation 26

5. Integral representation for the particle 27 displacement

6. Integral equation formulation of 31 elastodynamic diffraction problems

References 36 Appendix A. Calculation of the Green's

solution 37 Chapter II

FAR-FIELD RADIATION CHARACTERISTICS OF ELASTIC

WAVES AND THE ELASTODYNAMIC RADIATION CONDITION 41 42

43 1. Introduction

2. Integral representation for the particle displacement

3. The elastodynamic radiation condition 44 (physical considerations)

A. The mathematical formulation of the 48 elastodynamic radiation condition

5. The far-field radiation pattern 51

6. The power flow across S. 53 7. Conclusion

53

References -= Appendix eg

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Chapter III

A THEOREM ON THE SCATTERING AND THE ABSORPTION CROSS-SECTION FOR SCATTERING OF PLANE, TIME--HARMONIC, ELASTIC WAVES

1. Introduction

incident waves.

3. The cross-section theorem References

Chapter V

DIFFRACTION OF TIME-HARMONIC ELASTIC WAVES BY A CYLINDRICAL OBSTACLE

59

60

2. Description of the configuration . g. Power flow intensities of the

66 69 Chapter IV

RECIPROCITY RELATIONS FOR SCATTERING OF 71 PLANE, ELASTIC WAVES

1. Introduction yo 2. The elastodynamic field outside the __

obstacle

3. The elastodynamic field inside the _r obstacle

4. The reciprocity relations for the 77 far-zone scattered field

References 80

81

1 . Introduction

2. Description of the configuration and reduction of the three-dimensional theory to the two-dimensional one

3. Basic relations for two-dimensional disturbances

4. Exchange of acoustic energy

5. Elastodynamic reciprocity relation

6. Integral representation for the particle 90 displacement 82 82 85 88 90

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7. Diffraction of two-dimensional elastic

waves by an obstacle embedded in an

elastic medium of infinite extent

8. The far-zone approximation of the

scattered field

9. Numerical solution of the coupled integral

equations

10. Numerical results

1. Introduction

2. Description of the configuration. Integral

representation for the particle

displace-ment of the scattered wave

3. Integral equation for the jump in the

traction across the strip

4. The far-zone approximation of the

scattered field

5. Numerical solution of the

integral-93

98

104

107

References 135

Appendix A 136

Appendix B 138

Chapter VI

SCATTERING OF PLANE, ELASTIC WAVES BY A PLANE,

RIGID STRIP

141

142

143

147

148

-equation

.^^

6. Numerical results

151

156

References

Chapter VII

SCATTERING OF PLANE, ELASTIC WAVES BY A PLANE

CRACK OF FINITE WIDTH 159

1. Introduction ,

160

2. Description of the configuration.

Integral representation for the particle

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3. Integral equation for the jump in the particle displacement across the crack 4. The far-zone approximation of the scattered

field

5. Numerical solution of the integral--equation 6. Numerical results References 165 168 169 171 178 SAMENVATTING ,7g LEVENSBERICHT ,32

'This part has been reprinted from Applied Scientific Research _31_(1975)p.29-p.51 ,in which it has been published.

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Summary

The subject of investigation of the present thesis is the linearized theory of diffraction of time-harmonic, elastic waves by obstacles of finite extent. The main tool in the analysis is the integral-equation formulation of the problem. The integral equation is to be satisfied by the particle displacement on and/or the traction across the boundary of the obstacle. The general formulation of the problem is the subject of investigation of Chapter I. The derivation of the integral equation is based upon the integral representation for the particle displacement in an elastic medium. This integral representation follows from the Betti-Rayleigh reciprocity theorem, when in the latter a suitable

Green's state is subtituted for one of the elastodynamic states occurring in it. By letting the point of observation approach the boundary of the obstacle, an integral equation results. This integral equation applies to scattering by an obstacle of a rather general kind: it may be either a void, a perfectly rigid body, a body with elastic properties differing from those of its environment or a combination of these.

Another theoretical aspect of the diffraction problem at hand is the condition to be imposed on the scattered field at infinity, i.e. at large distances from the obstacle. In diffraction theory, this condition is known as the "radiation condition". In Chapter II, its mathematical formulation and the underlying physical considerations, both pertaining to elastic waves, are discussed. From the physical point of view, it is argued that the time-averaged power radiated away from a source of finite extent (in a scattering problem, the obstacle can be considered as the source exciting the scattered wave) is finite and positive. This condition implies that at large distances from the source, the particle displacement and the stress are inversely proportional to the distance from the source, while the resulting spherical wave should propagate in the direction of increasing distance from the source. The angular distributions of the wave functions (particle displacement and stress) are called "radiation characteristics". The relations between the particle displacement and the radial traction are shown to be locally

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the same as applying to plane waves. Now, in a homogeneous, isotropic elastic medium of infinite extent, two kinds of plane waves exist independently, viz., the compressional or P-wave and the shear or S-wave. Accordingly, the radiation (scattering) characteristics can be separated into a P-wave part and an S-wave part. In the P-wave part the particle displacement is longitudinal with respect to the direction of propagation, whereas in the S-wave part the particle displacement is transverse with respect to the direction of propagation.

In Chapter III, it is shown that a close relation exists between the scattering characteristics and the total power extinguished (i.e. both scattered and absorbed) by the obstacle in case the incident wave is a plane wave. This theorem is known as the "cross-section theorem". It is shown that, in spite of the fact that in elastodynamic scattering the scattered P-wave as well as the scattered S-wave contribute to the time-averaged scattered power, the extinction cross-section appears to depend only on the value of that scattering characteristic that is of the same type (P or S) as the incident wave.

In Chapter IV, a reciprocity relation pertaining to the plane-wave scattering characteristics are derived. It is shown that the scattered field observed in the direction -3 and due to an incident plane wave in the direction ^ is related to the scattered field observed in the direction -B_ and due to an incident plane wave in the direction a_.

This reciprocity relation applies to two plane P-waves, two plane S-waves as well to one plane P-wave and one plane S-wave.

In Chapter V, the theory applying to the diffraction of two-dimen-sional wave motions is developed. In a two-dimentwo-dimen-sional wave motion, the geometrical configuration as well as the field quantities involved are assumed to be independent of one of the Cartesian coordinates. As such, the two-dimensional theory can be considered as a special case of the three-dimensional one. Following this line of reasoning, the two-dimen-sional version of the relevant three-dimentwo-dimen-sional relations are presented. As an application of the theory, the diffraction of plane elastic waves by a number of cylindrical obstacles is investigated. Numerical results for the extinction cross-section and for the normalized power scattering

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characteristics are presented.

In Chapter VI, the diffraction of plane elastic waves by a perfectly rigid plane strip is investigated. This leads to an integral equation in which the amount by which the traction jumps across the strip occurs as the unknown function. Numerical results for the extinction cross--section and for the normalized power scattering characteristics are presented.

Finally, in Chapter VII, the diffraction of plane elastic waves by a plane crack of finite width is investigated. In the integral equation the amount by which the particle displacement jumps across the crack occurs as the unknown function. Numerical results for the extinction cross-section and for the normalized power scattering characteristics are presented.

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SCATTERING OF ELASTIC WAVES

BY ELASTICALLY TRANSPARENT OBSTACLES

(INTEGRAL-EQUATION METHOD)*

T. H. TAN

Dept. of Elect Eng , Div of Electromagnetic Res., Delft Univ. of Technology, Delft, THE NETHERLANDS

Abstract

A formulation of elastodynamic diffraction problems for sinusoidaHy in time varying disturbances in a linearly elastic medium is presented Starting with the elastodynamic reciprocity relation, an integral representation for the particle displacement is derived In it, the particle displacement and the traction a t the boundary of the obstacle occur From the integral represen-tation, an associated integral equation is obtained by letting the point of observation approach the boundary of the obstacle The "obstacle" may be either a rigid body, a void, or a body with elastic properties differing from those of its environment, or a combination of these. The integral equation thus obtained is well-suited for numerical treatment, when obstacles up to a few wavelengths in maximum diameter are considered.

§ 1. Introduction

Many existing approaches to solving the problem of the diffraction

and scattering of acoustic waves by an obstacle in an elastic

me-dium are based upon the technique of separation of variables and

a corresponding expansion of the unknown vectorial particle

dis-placement in a series of associated transcendental functions. In this

category. White [1] considers the scattering, of both compressional

and shear waves, impinging obliquely on an infinitely long cylinder

of circular cross-section by expanding the unknown particle

dis-placement in a series of Hankel and Bessel functions. The

"obsta-cle" can be either a void, a rigid inclusion, or a cylinder with elastic

properties differing from those of its environment. Also, numerical

• The research reported m this paper has been supported by the Netherlands organi-zation for the advancement of pure research (Z.W.O ).

29

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results are presented for the cross-sections of to-shear,

shear-to-compressional, compressional-to-shear and

compressional-to-com-pressional wave scattering. Secondly, we want to mention the papers

by Ying and Truell [2], Einspruch and Truell [3], and Einspruch ,

Witterholt and Truell [4], who discuss the scattering, of both

com-pressional and shear waves, by a spherical obstacle, this being a

three-dimensional problem. In their papers, analytical results for

the scattering cross-sections are given in the form of expansions in

which the coefficients of the wave-functions expansions occur. Also,

the low-frequency approximations of these expansions are given.

Based upon [4], Kraft and Franzblau [5] give numerical results for

the scattering cross-section of a spherical cavity, whereas McBride

and Kraft [6] have calculated the scattering cross-section of an

elas-tic sphere embedded in an infinite elaselas-tic medium. A further

gener-dization of the technique has been given by Oien and Pao [7], who

consider the scattering of compressional waves by a mobile, rigid,

spheroidal obstacle; for axisymmetric excitation, this leads to a

scalar problem.

In the papers cited above, the separability of the elastodynamic

wave equation in a coordinate system that allows for the matching

of the boundary conditions, plays an important role. Although the

method yields, in principle, exact results, the appUcabiUty of the

technique is, however, severely limited, since the

elastodynamic-wave equation separates in two, orthogonal, curvilinear coordinate

systems only, viz. the circularly cylindrical and the spherical one [8].

To overcome this Hmitation, Banaugh [9] has formulated the

rele-vant boundary-value problem in terms of certain integral equations.

When these are solved, a set of surface potentials is obtained that

can be used to calculate the exterior field at any point in space. In

Banaugh's formulation, the particle displacement is written as the

sum of two contributions: a contribution due to compressional

waves in the form of the gradient of a scalar potential and a

con-tribution due to shear waves in the form of the curl of a vector

po-tential. In this way, Banaugh has obtained numerical values for

the surface potentials in the case of two-dimensional rigid cylinders

of arbitrary cross-section. By emplo5nng the method of integral

equations, Banaugh has indeed eUminated the limitations of the

method of separation of variables. However, the formulation of the

boundary conditions to be imposed on the surface potentials, is

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cumbersome. Moreover, the surface potentials themselves do not

correspond directly with elastodynamic physical quantities.

In the present paper we present, for sinusoidaHy in time varying

disturbances in a reciprocal elastic medium, an alternative approach

to the formulation of elastodynamic diffraction problems. Starting

with the elastodynamic reciprocity relation, we derive, with the aid

of a suitable auxiliary (or Green's) elastodynamic state, an integral

representation for the particle displacement, expressed in terms of

the particle displacement and the traction at the boundary of the

obstacle. From the integral representation, an associated integral

equation is obtained by letting the point of observation approach

the boundary of the obstacle. For the special case of an isotropic

medium, the resulting integral equation is similar to the one derived

by Kupradze [10].

The "obstacle" maybe either a rigid body, a void, or a body with

elastic properties differing from those of its environment, or a

com-bination of these. For the special type of obstacle in the form of a

semi-infinite crack or a semi-infinite baffle, the integral-equation

method has been used by Maue [11] for time-harmonic waves and

by de Hoop [12] for impulsive waves (see also Achenbach [13]).

Finally, we want to mention the review paper on elastic wave

prop-agation problems by Miklowitz [14].

In Section 2, a summary of the basic relations in the theory of

acoustic waves in an elastic medium is presented. The discussion on

the exchange of acoustic energy is the subject of Section 3. In

Sec-tion 4, the elastodynamic reciprocity relaSec-tion serves as a point of

departure in the derivation of an integral representation for the

particle displacement; this procedure will be discussed in Section 5.

In Section 6, starting with the representation theorem for the

par-ticle displacement, the associated integral equation is derived.

Fi-nally, in the Appendix, it is shown how, in principle, the auxiliary

(or Green's) elastodynamic state, required in the integral

represen-tation for the particle displacement, can be calculated. For an

iso-tropic elastic medium, the Green's elastodynamic state is obtained

explicitly.

§ 2. Basic relations in the theory of elastic waves

The main object of this section is to summarize the basic relations in

the theory of acoustic waves in an elastic sohd, occupying a finite or

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infinite domain in three-dimensional space. In view of the

applica-tions that we have in mind, we restrict our analysis to wave moapplica-tions

of sufficiently small amphtude, i.e. the ampUtudes are so small that

it is allowed to employ the linearized version of the general relations.

For a general discussion on the subject we refer to Mason [15] and

Love [16]. We assume that all field quantities vary sinusoidally in

time with circular frequency co. Unless stated exphcitly otherwise.

the complex representation of the field quantities is used, according

to the rule that a certain field quantity as a function of time is

ob-tained by taking the real part of the product of its associated

com-plex representation and the comcom-plex time factor exp{—i(ot), where

i is the imaginary unit and t denotes the time. In the formulas, the

complex time factor will be suppressed. To locate a point in space,

we use Cartesian coordinates xi, %% and xz, which in this order form

a right-handed system. Throughout the calculations, latin

scripts are to be assigned the values 1, 2 and 3 ; for repeated

sub-scripts the summation convention holds. Occasionally, bold-face

type letters will be used to denote vectorial quantities; in

particu-lar X will denote the position vector. Sl-units are used throughout.

We start with the linearized equation of motion. This can be

written as [17],

% ! . ; + pw^Mi = —/i, (2.1)

where

Sy = partial derivative with respect to x^ (in m~i),

Tf,y = (complex) stress (in N/m^),

p = volume density of mass (in kg/m^),

to = circular frequency (in rad/s),

Mj = (complex) particle displacement (in m),

/i = (complex) volume density of body force (in N/m^).

In (2.1), the quantities p = p(*), Ui = Ui(x) and /j = /i(*) are

as-sumed to be piecewise continuous functions of position, while

T j j = Ti,i[x) is assumed to be a piecewise continuously

differen-tiable function of position. The elastic properties of the medium

are accounted for by the (linearized) constitutive relation, which

expresses the stress linearly in terms of the strain. The linearized

expression for the latter quantity is given by [18]

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The constitutive equation is written as

Ti,; = c j j , 3,,j,ej,,g, (2.3)

in which C(,y,j,,j = Cij,p^q{x, u>) are the stiffness coefficients in the

frequency domain (in N/m^). A medium is called perfectly ela^stic

if the stress and the strain interact instantaneously. The stiffness

coefficients are then real and independent of frequency. However,

also linear visco-elastic effects can be taken into account by

con-sidering complex-valued and frequency-dependent stiffness

coef-ficients. In the time domain, the latter lead to a constitutive

equa-tion of the form

Ti.}(x. t) = j r VU.P.dix. n Ep,q(x, t - t') At', (2.4)

in which Ttj{x, t) denotes the real stress, Ep,q{x, t) the real strain

and »yj,y, j),«(*, i) the visco-elastic time response. The latter quantity

is related to Cij^p^g through

cu,p,t{x, w) = J~ exp(io>r) jyi,y,p,g(*, t') di'. (2.5)

The Umits of integration in (2.4) and (2.5) take into account t h a t

the elastic behaviour of a medium is causal. The type of constitutive

relation (2.4) has been introduced by Boltzmann [19].

Since both the stress and the strain are symmetrical tensors, the

stiffness coefficients satisfy the symmetry relations

Cij,p.g{x, «) = Cij,g,p(x. ft)) = cy,i,j,,a(*, (o) = Cj,t,g,p(x, co).

(2.6)

On account of (2.6) the number of independent stiffness coefficients

reduces to 36. How many coefficients actually differ in value,

pends on the crystallographic properties of the material. For

de-tails on this subject we refer to Nye [20], who also gives some

nu-merical values of the stiffness coefficients for different materials.

For an isotropic material, the elastic properties are independent of

the orientation of the axes of the Cartesian reference frame; then

the number of different stiffness coefficients reduces to 2 and we

have

^tJ.P.Q = ^^i,]^p,q + M^i,P^J,Q + ^l,Q^],p)' (2-7)

where A = A(*, co) and ft = /i{x, co) are the Lam^ coefficients of the

material in the frequency domain. In a domain where the medium

is homogeneous, p and Cij^p^g are independent of position.

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In several appUcations, parts of the elastic medium do not

sist of a solid, but of a fluid material. In the latter case, we

con-sider the fluid to be perfect in the sense that internal friction is

ab-sent, which leaves the compressibihty to be taken into account. The

stress tensor then reduces to the diagonal form

"Tij = —PKj.

(2.8)

where p = p[x) denotes the scalar pressure. Further, the

consti-tutive relations (2.3) are to be replaced by

f = -KdqUg,

(2.9)

in which K = K(x) denotes the modulus of compression of the fluid.

At those locations where a discontinuity in properties of the

me-dium occurs, the equation of motion (2.1) no longer appUes, since

at least some of the derivatives occurring in it do not exist. At

those locations the equation of motion has to be supplemented by

boundary conditions. To this aim we assume that the volume

den-sity of mass and/or the stiffness coefficients at most jump by finite

amounts when crossing a surface of discontinuity in properties. We

further assume that no rupture of the material takes place. Let 27

be a two-sided surface of discontinuity. Further, let n denote the

unit vector along the normal to Z.

A summary of the boundary conditions at Z is given in Table I.

TABLE I

Elastodynamic boundary conditions

Type of boundary Particle displacement «( Traction T(,/n^ firm contact

rigid boundary stress free boundary solid/fluid interface

continuous across the interface vanishes on the boundary unspecified on the boundary normal component continuous across the interface, tangential components unspecified

continuous across the interface unspecified on the boundary vanishes on the boundary normal component continuous across the interface, tangential components vanish on the interface

§ 3. Exchange of acoustic energy

The starting point for our considerations on the exchange of

acous-tic energy due to the presence of wave motions in an elasacous-tic

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mate-rial, is the expression for the time rate at which work is done by the

body forces. This quantity is given by the product of the volume

density of the (real) body force and the local value of the (real)

particle velocity. Since both quantities vary sinusoidally in time

with circular frequency o), their product contains a

time-indepen-dent part and a part that varies in time with circular frequency 2a).

Since in most appUcations the time averages of the acoustic power

flow and of the stored energy are the important quantities, we shall

proceed our discussion by considering these quantities. Expressed

in terms of the complex representations we have

Re[/i(«) exp(—icoi)] Re[2;i(«) exp(—ico^)] =

= i(/i< + f:n) + kihn exp{-2icot) + / > ; exp(2ico^)], (3.1)

where * denotes the complex conjugate and Vi denotes the (complex)

particle velocity. Let

< . . . > r ^ r - i | l : + = ^ . . . d i (3.2)

denote the time average over a single period T = 27t/(y in time.

Then it follows from (3.1) that

<Re[/«(*) exp(-ico<)] Re[vi{x) exp(-ift>0]>T = J Re[/4(«) v^{x)]. (3.3)

From the Hnearized equation of motion (2.1) we have

fiVi = —Vi^jTUj — i(opViv;. (3.4)

Using the identity

•"i^inj = SjiViru) - njdjv^i

= ^jiK'^i,]) — ^(^ij^h' (3-5)

we can rewrite (3.4) as

fiV* = —S;(w,*Ti,;) -f- icoTtjelj — icopVtV^. (3.6)

In order to interpret the different terms in (3.6) we integrate the

equation over a bounded domain i^. Let y be the boundary of TT

and let n denote the unit vector along the outward normal to y

(see Fig. 1). Then an integration of (3.6) over TT and applicatioii of

Gauss' divergence theorem yields

ijir fiVi AV = - §y v^Tijfij dA - ift) JJj^ pViV^ dV

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Fig. 1. Domain for which the complex acoustic power balance is derived.

The various terms in (3.7) can be interpreted as follows

P = §^ (^—:^v*Ti,j) fij dA = complex outward acoustic

power flow through 6^, (3.8)

I^kin = i JJjy p^iV* dV = time-averaged kinetic energy

stored in TT, (3.9)

Waet = i ni*' '''iJ^i.i ^ ^ ~ complex elastic deformation

energy stored in y , (3.10)

•Pext = i jjj-r /i^,* dV = complex power delivered by the

body forces in 'V. (3-11)

For the time-averaged values of each of these quantities we have

to take the real part of the corresponding complex expression. With

the aid of the quantities introduced in (3.8)-(3.11), (3.7) can be

written as

Pext = P - 2ico(l^kin - Wiet); (3.12)

this equation is known as the complex power balance. In connection

with the exchange of acoustic energy we distinguish the following

properties of the material present in TT :

(a) The material is passive if

Re[Pext - - P ] > 0. (3.13)

Since IFkin is real, (3.13) implies that

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for any domain "f and any state of disturbance of the medium.

Con-sequently (3.13) implies t h a t

Im[Tije;_y] < 0 (3.15)

at any point of the material. When the medium is isotropic, the

inequahty (3.15) can be written as

lm[XdpUpdiU* + IfidjUtdjU*] < 0 (3.16)

which implies t h a t

I m [ A ] < 0 and Im[u] < 0 (3.17)

at any point of the material.

(b) The material is lossless if

Re[Pext - P] = 0. (3.18)

Now, (3.18) implies that

lm[ruelf] = 0 (3.19)

for any domain "f and any state of disturbance of the medium.

Consequently, we have

Im[Ti,y<,-] = 0 (3.20)

at any point of the material. When the medium is isotropic, the

equality (3.20) can be written as

lm[X8pUpdiU* + IfidjUidjU*] = 0 (3.21)

which implies that

Im[A] = 0 and Im^a] = 0 (3.22)

at any point of the material.

The vector quantity

S) = - i < T « . ; (3.23)

occurring in (3.8) represents the complex acoustic power flow

den-sity. This quantity can be considered as the elastodynamic

counter-part of the Poynting vector in electromagnetic wave theory. I t is

noted, that in a fluid 5 ; is given by S; = ^v^f. On account of the

conditions imposed on the elastodynamic quantities when crossing

a surface of discontinuity we have (see Table I),

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i.e. the normal component of the acoustical power flow density is

continuous when crossing a surface of discontinuity. Further, we

have for the special cases of the boundary conditions,

Sj = 0 at a rigid boundary, (3.25)

SjHj = 0 at a traction-free boundary, (3.26)

and

Sjttj continuous across a solid/fluid interface. (3.27)

§ 4. Elastodynamic reciprocity relation

The elastodynamic reciprocity relation, also referred to as the

Betti-Rayleigh theorem, expresses a relation between elastodynamic

quantities in two possible, but different, states. The theorem can be

considered as the elastodynamic counterpart of Lorentz's

reciproc-ity relation [21] pertaining to time-harmonic electromagnetic fields.

The time-dependent form of the elastodynamic reciprocity theorem

for states with quiescent past has been discussed by Wheeler and

Sternberg [22], whereas Kupradze [23] has considered the

reciproc-ity relation for lime-harmonic elastodynamic fields. In both papers,

the medium under consideration is assumed to be isotropic. A

cor-responding reciprocity relation pertaining to piezo-electric media,

which are anisotropic, has been derived by De Jong [24]. A

time-dependent (with arbitrary time dependence) reciprocity relation for

linear and causal, but otherwise general (i.e. anisotropic and

visco-elastic) media has been given by De Hoop [25]. In the present

sec-tion, we briefly review the derivation of the reciprocity theorem for

time-harmonic elastodynamic fields. The medium under

consider-ation is assumed to be linear and reciprocal; since the stiffness

coef-ficients may be complex, visco-elastic materials are included in the

discussion. The reciprocity condition implies t h a t the following

symmetry relation for the stiffness tensor holds

Ci,],p,q{X, U)) = Cp,qjj{x, co) at all *. (4.1)

This shows that, in particular, an isotropic medium is reciprocal,

since the right-hand side of (2.7) satisfies (4.1).

Let the two elastodynamic states be distinguished by the

super-scripts A and B, respectively. Both elastodynamic states are

pre-sent in one and the same medium and vary sinusoidally in time with

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the same circular frequency (o. The equations of motion and the

constitutive equations pertaining to both states are of the form

(2.1) and (2.3), respectively.

We now consider the expression for ffuf — ffuf and ootain by

using (2.1)

/ . V - ff»? = <^i<i - <h<i- (4-2)

On account of (4.1) we have

TiV/wf - -^Ifii^i = 0- (4-3)

Consequently, (4.2) can be written as

ffuf - ffuf = d^iufrf, - ufrfj. (4.4)

Let y be a bounded, sufficiently regular, closed surface and let i^

be its interior. Integrating (4.4) over i^ and using the divergence

theorem we obtain

iii^ ( A V - fM) ^V = §y {ufrfj - utrl,) Hi dA, (4.5)

in which n denotes the unit vector along the outward normal to 5".

In the next section, the reciprocity relation (4.5) will be used to

derive certain integral representations for the particle displacement.

The relevant integral representations are the elastodynamic

counter-parts of Green's third identity in potential theory and play a vital

role in the integral-equation formulation of elastodynamic

diffrac-tion problems.

§ 5. Integral representation for the particle displacement

In the present section we derive an integral representation for the

particle displacement M<, associated with a sinusoidally in time

vary-ing disturbance in a reciprocal elastic medium. To this aim we

in-troduce an auxiliary state of disturbance, a called Green's

so-lution; for the moment it is defined as being any vector-valued

displacement field and its associated tensor-valued stress field that

are due to a localized force that operates at a single point 0* with

position vector x^. Denoting the corresponding quantities by the

superscript G, we have ff = ai6{x — x^), where d(x — x^) denotes

the three-dimensional unit pulse and aj is an arbitrary constant

vector. The equation of motion and the constitutive equation

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per-taining to the Green's solution are

djrfjx, x^) + pw^u9(x, x^) = —aid{x — * , ) , (5.1)

and

T u ( * . *#) = Ci,j,p,q{X, Oi) 8pU^(x, x^). (5.2)

The acoustic field for which an integral representation is to be

ob-tained, satisfies the equation of motion (2.1) and the constitutive

relation (2.3).

We now apply the reciprocity relation to a domain •V, bounded

by a sufficiently regular, bounded, closed surface £^. Further, we

take the two states in (4.5) as follows: {uf, rfj) = {ut, TIJ} and

{"f' '^fj) — {"?• '^?j}- Using the property of the three-dimensional

unit pulse, we obtain

- f^ (wi^u - '^f-'u) njdA+ jjjy fiuf dV =

0 when ^ e f^,

aiUi{x^) when 0> eV, (5.3)

where V^ is the domain outside y . From (5.3), a representation for

the particle displacement is obtained by observing that a* is

arbi-trary and that the Green's solution depends linearly on the

com-ponents of a*. To express this dependence we introduce a tensor of

rank two

M^^

and a tensor of rank three

T?,-.^

which relate uf and

TJ^; to afc through

«,^ = < * « * (5-4)

and

Substituting (5.4) and (5.5) in (5.3), and noting that the resulting

equation should hold for any choice of a*, we obtain

- f5' (r5;fcMi - w.^*r,.y) ny d ^ + j j j ^ ulji dV =

0 when ^ e TT^, (5.6)

«fc(«ai) when ^ 6 TT.

Our subsequent analysis also requires the values of the different

terms in (5.3) when 9 is chosen on the boundary surface 9'. To this

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Fig. 2. Integral representation for the particle displacement when ^ lies on the surface £f.

aim we consider the configuration shown in Fig. 2. We draw a sphere

of radius e around 0* and call Zc t h a t part of 9 that is located inside

it, 9'e. that part of the surface of the sphere that is located inside S/",

and •Ve, the common part of the interior of the sphere and •f. Taking

the two states as before, application of the reciprocity relation (4.5)

to the domain y — "Ve with boundary surface £/' — Ze -\- S^e yields

- fy-^.+i., (mrl^ - ^hui) % d ^ + \\\r-r. h^f ^V = 0, (5.7)

since 0* is located outside the domain to which the reciprocity

rela-tion is applied. As in the case of the scalar Green theorem, the

contribution from the various terms in (5.7) reduces in the limit

e ^ 0 to

l\y, ^trfjfi] dA -> \aiUi{x^) as e -> 0,

lly. Mf T/,yW/ dA -^0 as e -^ 0,

\\\y-^^...dV^\\\^...dV as £ ^ 0 ,

\\^-z^ ...dA ^l^y ...dA as e -^ 0,

where it is understood that in the neighbourhood of ^ the surface

is locally flat and consequently has at ^ a unique tangent plane.

Further, ff denotes the Cauchy principal value of the relevant

sur-face integral. The proof of these relations rests upon a local

expan-sion of uf and rfp which are of the orders 0(e~i) and

0{E~'^),

re-spectively. I n the case of an isotropic medium, details are given by

Kupradze [26]. Collecting the results, (5.7) reduces in the Hmit

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e - > 0 to

- ijy {uirf,i - ufrt,]) fij dA + JjJ^ fiuf dV = ^UiUiix^)

when 0e9. (5.8)

Substituting (5.4) and (5.5) in (5.8) and noting that the resulting

equation should hold for any choice of a*, we obtain

- ffy {•'u-.k^i - w."fcT<,y) nj dA + JjJ^ uf.Ji dV = ^Uk{x^)

when ^ei^. (5.9)

The different representations derived thus far are restricted to

bounded domains. In order to obtain the corresponding

represen-tation for unbounded domains, we consider the configuration shown

in Fig. 3. Let iT be a bounded domain and let y be its boundary

surface. Let 6^A be a sphere with radius A around 0' and let f^t

denote the domain interior to 6^A ', ^ is chosen so large that 9A

Fig. 3. Integral representation for the particle displacement in the unbounded domain outside 6^.

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completely surrounds 9. Taking the two states as before,

applica-tion of the reciprocity relaapplica-tion (4.5) to the domain •f A — "f yields,

IFy (T?,M< — wfrij) n.) dA — §y^ (MjT.^y — u^Tij) rij dA +

0 when 0>Ei^, (5.10)

+ my.-rfiU?dV =

aiUi{x^) when ^ 6 T^J — iT.

If, furthermore the Green's solution satisfying (5.1) and (5.2)

con-sists of waves diverging from the source point ^ and the disturbance

{wj, Tij) satisfies the so-called elastodynamic radiation condition, the

surface integral over 9A vanishes in the limit -d -> oo. The

vanish-ing of this surface integral can for the moment be regarded as the

necessary and sufficient condition for expressing, be it imphcitly,

the elastodynamic radiation condition. How this condition is

re-lated to conditions to be imposed on [ui, rtj], will be discussed in a

separate paper [27]. Again, taking into account the linear

depen-dence of rfj and uf on a/c, we arrive at

§y (^U:*w« - »?;kruj) nj dA + \\\y„ fiU%^ dV =

0 when ^ e TT, (5.11)

U)c{x^) when 0>e'r^,

where n denotes the unit vector along the normal to 9 as indicated

in Fig. 3 and •f ^ = hmj^^o VA — i^. For determining the values

of the various terms occurring in (5.10) when 0* is chosen on the

boundary surface 9, we employ the same procedure as for a bounded

domain. The result for a locally flat surface is

when 0'e9. (5.12)

The integral representations of the type (5.6), (5.9), (5.11) and (5.12)

play an important role in the integral-equation formulation of

elastodynamic diffraction problems.

§ 6. Integral equation formulation of elastodynamic diffraction problems

In this section we consider the diffraction of elastic waves by

ob-stacles of a rather general kind and in particular the formulation of

these diffraction problems in terms of integral equations. The

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con-Fig. 4. The composite obstacle.

figuration to be investigated is shown in Fig. 4. In a homogeneous,

elastic medium of infinite extent, an obstacle of bounded elastic

contrast is present. The obstacle occupies the domain inside a

suf-ficiently regular, bounded, closed surface ^^C). The unbounded

do-main outside y*") is called iTW. The dodo-main occupied by the

ob-stacle consists of a finite number A^ of sub-domains T^'P) {p = I,

..., N). The boundary of i^<P> is the closed surface

9<P'>;

the latter

is assumed to be sufficiently regular. In the analysis, the common

part of 9<'P^ and 5^(9) will be denoted by

9^P'«1

For notational

consistency, 9'-^^ itself will also be denoted by

9<V,P).

With the

convention that, in the summation, empty intersections are to be

omitted, we can write

N

9(v) = ^ c^iP.d) {p = 0,...,N). (6.1)

g=0 Q*P

The unit vector along the normal to 9^P^ will be denoted by n^P)

(p = \,2, ..., N) and is chosen such that it points away from i^(p>.

One exception to this convention will be made: the unit normal

nC) to 9<0) points toward TTW.

In each of the domains Tr<P>, sources of elastodynamic radiation

may be present; they are all assumed to vary sinusoidally in time

with the same circular frequency co. The operation of the sources is

accounted for by a prescribed volume density of body force / = f^P'>

when * e i^(p). The corresponding elastodynamic state is denoted by

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and the stiffness coefficients and the mass density by

K / . p , ? . p } = {clf/.p,,,p(^'} when xer(v). (6.3)

The right-hand sides of (6.2) and (6.3) are to be substituted in the

equation of motion (2.1) and the constitutive equation (2.3) when

apphed to the domain iTfJ').

The diffraction problem we want to solve amounts to calculating

the total elastodynamic wave motion that is generated by t h e

sources in the heterogeneous configuration under consideration. I n

this respect, it will be advantageous to treat the elastodynamic

wave motion in the unbounded domain y^") somewhat differently

in order to allow for the case of the incidence of a plane wave. To

this aim, we define the "incident field" \u™,

TJ-°},

as the

elasto-dynamic state that would be generated by the sources in T^C' in

the absence of any other sources as well as of the inhomogeneities

introduced by the obstacle. I t is remarked that M^° and rj° are

de-fined in the entire three-dimensional space and satisfy the equation

of motion

a;rl°, + p(0)coy.'^ = -/<«) (6.4)

and the constitutive relation

Next, we introduce, in •fC^l, the scattered field wf and T^J as

W|« def ^m _ ^In ^jjgj^ ^ £ ^(0), (6.6)

T^^.^rf]-T^^ when xe-TW. (6.7)

From (6.4)-(6.7) we conclude that the scattered field has to satisfy

the equation of motion

^rZ + /''"'w^wf = 0 when * G -r«'), (6.8)

and the constitutive equation

^j ='^u.P.A^ when xei-m. (6.9)

As to the behaviour of the scattered field at infinity, we require

that is satisfies the elastodynamic radiation condition mentioned in

Section 5. The first step towards the formulation of the diffraction

problems in terms of integral equations consists of deriving proper

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integral representations for M^'* {q = \,2, ..., N). Let the Green's

state for an unbounded domain with the properties of y^(9> be

de-noted by {uf_^^\x, x^), rf_^,^l{x, x^)}, then the desired representation

for M^«' is obtained from an apphcation of the representation

theo-rem (5.6) and (5.9) to the domain ':^<«> with q ^ 0. The result is

- f^<.. {<^%{^. *^) uf{x) - uff{x. x^) T<«>(*)) nf dA +

«1?*(*^) when ^£1^(9),

+ lll^.>/<«H*Xf(*,*^)d^ =

J4''(*#) when ^e5^(9), (6 10)

0 when 0'ei^^£.

In (6.10), it is understood that the surface integral over 5^(9) has to

be interpreted as a Cauchy principal value whenever 0 hes on y<9).

A representation for M*"^ is obtained in a somewhat different manner.

We first apply the representation theorem to the field {MJ", T^^} and

to the domain bounded internally by 5^(0) and externally by a sphere

9A of radius A around &. By virtue of the elastodynamic radiation

condition imposed on the scattered field, the surface integral over

9A vanishes in the limit zl ^- oo. Hence,

|f^<o. (Te)?i(*. *^) <"(*) - <<."'(*. **) <"(«)) ^T d^ =

<"(*#) when 0 e TTW),

>f(«3.) when 0e9^°\ (6 11)

0 when 0 e interior of 5^0).

In addition, application of the representation theorem to the field

{M;°, TJ^J and to the interior of 9^^^ yields

- §^-.0. {r?j%{^. *^) < ( * ) - <'."'(*. *^) r5(*)) < ^ d^ =

0 when 0> e r^^\

K°(*#) when ^ e y ( O ) , (6.12)

<"(*#) when ^einterior of ^(0).

Combimng (6.11) and (6.12) we obtain

f ^<. (rej?i(*, *^) "fH*) - < r ( * . *^) rl!K*)) < * d^ =

u^^{x^) when

^ G - T C ) ,

K°'(«#) - <(*a') when ^G.^(0), (6.13)

—<"(*#) when ^ G interior of 5^(0).

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From (6.10) and (6.13), a system of coupled integral equations is

obtained by choosing the point of observation on 9^1^ and taking

into account the appropriate boundary conditions. Assuming that

the bodies are in firm contact, these boundary conditions are (cf.

Table I)

u^\x)= uf{x) =,^(^'9)(«) when x & 9(v^^\ (6.14)

rf){x)nf'' = rf^(x)nf^ = %f^^'''\x) when xe9(v,<i). (6.15)

With the aid of (6.1), (6.14) and (6.15), (6.10) and (6.13) together

can be written as

(2^..o - 1) S n^<... (r?)n(*, x;) nfj>f'^\x) +

r=0 r*(l

- uf^i\x, x^) f<^-^\x)) dA +

+ (1 - '5,.o) m^«. /<•«'(*) t^?^\x. *^) dF = u ' r ' M +

- Sq,ou^kM when 0 G 9'-"' »>

with q = 0, ...,N, s = 0, ...,N; s ^q. (6.16)

The integral equations (6.16) constitute a system of coupled integral

equations, from which for a given excitation the particle

displace-ment and the normal components of the stresses on the interfaces

can be calculated. With the aid of the representation theorems (6.10)

and (6.13), the particle displacement in the entire three-dimensional

space can be obtained. Subsequent application of the constitutive

equation then gives the value of the stress at any point in space.

For obstacles of reasonable shape and of dimensions not too large

compared with the wavelength, this system can be solved

numeri-cally, provided that not too complicated expressions for the Green's

state {uf.,^{x, «#), Tfj.^.{x, x^)} can be derived. In practice, the latter

condition makes it necessary to concentrate ourselves for the

mo-ment on obstacles that are piecewise homogeneous .and isotropic.

In the Appendix it is shown how, in principle, for an arbitrary

anisotropic, but homogeneous medium, the Green's state for an

un-bounded domain can be expressed as a spatial Fourier spectral

rep-resentation. For an isotropic medium, simpler expressions result,

and this enables us to use them in the numerical procedure. The

numerical evaluation of the spatial Fourier integrals at present still

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meets with difficulties that have not been surmounted, except when

apphed to the relatively simple configurations (see Neerhoff [28]).

We conclude this general integral-equation formulation of our

diffraction problem with an indication which modifications have to

be made in (6.16) in order to include the special types of

"obsta-cles", the boundary conditions of which we have indicated in

Table I. It is remarked that in the domain occupied by either a

void or a perfectly rigid obstacle, the particle displacement (and the

stress) can be put equal to zero identically. This imphes that in

(6.16), for those domains we have to put /J*' and {M?^^ '^?,ifk} equal

to zero.

Acknowledgment

The author wishes to thank Professor A. T. de Hoop for his

sugges-tions and remarks. The financial support of the Netherlands

orga-nization for the advancement of pure research (Z.W.O.) is

grate-fully acknowledged.

Received 4 March 1975

R E F E R E N C E S

[1] W H I T E , R . M . , J . Acoust. Soc. Am. 30 (1958) 7 7 1 .

[2] YiNG, C. F . a n d R H O N T R U E L L , J . Appl. P h y s . 27 (1956) 1086. [3] E I N S P R U C H , N . G . a n d R H O N T R U E L L , J . A c o u s t . Soc. A m . 32 (1960) 214. [4] E I N S P R U C H , N . G . , E . J . W I T T E R H O L T a n d R H O N T R U E L L , J . A p p l . P h y s . 31 (1960) 806. [5] K R A F T , D . W . a n d M. C. F R A N Z B L A U , J . Appl. P h y s . 42 (1971) 3019. [6] M C B R I D E , R . J . a n d D . W . K R A F T , J . Appl. P h y s . 4 3 (1972) 4853. [7] O I E N , M . A. a n d Y . H . P A O , T r a n s . ASME, Ser. E 40 (1973) 1073.

[8] M O R S E , P . M. a n d H . F E S H B A C H , Methods in Theoretical Physics, McGraw-Hill Book C o m p a n y , N . Y . , 1953, p . 1759-1791.

[9] B A N A U G H , R . P . , S c a t t e r i n g of Elastic W a v e s b y Surfaces o j A r b i t r a r y S h a p e , Thesis, University of California, Livermore, California (1962).

See also B A N A U G H , R . P . and W . G O L D S M I T H , T r a n s . A S M E , Ser. E 30 (1963) 589. [10] K U P R A D Z E , V . D . , Progress in Solid Mechanics, Vol. I l l , ed. b y I. N . S N E D D O N a n d

R. H I L L , N o r t h - H o l l a n d P u b l . Co., A m s t e r d a m , 1963. [11] M A U E , A . - W . , Z . Angew. M a t h . & Mech. 3 3 (1953) 1.

[12] D E H O O P , A. T . , R e p r e s e n t a t i o n theorems for t h e displacement in an elastic solid a n d their application t o elastodynamic diffraction t h e o r y . Thesis, Delft U n i v e r s i t y of Technology, t h e N e t h e r l a n d s (1958) i 22 .

[13] A C H E N B A C H , J . D . , W a v e P r o p a g a t i o n in Elastic Sohds, N o r t h - H o l l a n d P u b l . Co., A m s t e r d a m , 1973, Ch. I X .

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[15] M A S O N , W . P . , E d . , Physical Acoustics, Vol. I * , Academic Press, New Y o r k L o n d o n , 1964, p . 2 - 1 0 9 .

[16] L O V E , A. E . H . , A T r e a t i s e on t h e M a t h e m a t i c a l T h e o r y of E l a s t i c i t y , Dover Publi-c a t i o n s , N . Y . , 1944, 4 t h . ed., Ch. X I I I .

[17] A U L D , B . A., Acoustic Fields a n d W a v e s in Solids, Vol. I, J o h n - W i l e y & Sons, New Y o r k , 1973, p . 45.

[18] A U L D , B . A., loc. cit. p . 11.

[19] BOLTZMANN, L . , A n n . der Physik u n d Chemie. E r g a n z u n g s b a n d 7 (1876) 624. [20] N Y E , J . F . , Physical P r o p e r t i e s of Crystals, Clarendon Press, Oxford, 1957, Ch. V I I I . [21] LoRENTZ, H . A . , Versl. K o n . N e d . A k a d . W e t . A m s t . 4 (1896) 176.

[22] W H E E L E R , L . T . a n d E . S T E R N B E R G , Arch. R a t . Mech. a n d Analysis 31 (1968) 5 1 .

[23] K U P R A D Z E , V. D., loc. cit. p . 9.

[24] D E J O N G , G . , G e n e r a t i o n of Acoustic W a v e s i n Piezo-electric devices. Thesis, Delft U n i v e r s i t y of Technology, t h e N e t h e r l a n d s (1973). See also: D E J O N G , G . , Appl. Sci. Res. 26 (1972) 445.

[25] D E H O O P , A. T., A p p l . Sci. Res. 16 (1966) 39.

[26] K U P R A D Z E , V. D . , loc. cit. p . 26.

[27] T A N , T . H . , to be published in Appl. Sci. Res.

[28] N E E R H O F F , F . L . , to be published in P r o c . R. Soc. Lond. ser. A 342 (1975) 237. [29] T I T C H M A R C H , E . C , I n t r o d u c t i o n to t h e T h e o r y of Fourier I n t e g r a l s , Clarendon

Press, Oxford, 1948, p . 42.

[30] E A S O N , G . , J . F U L T O N a n d I. N . S N E D D O N , Phil. T r a n s . A, 248 (1956) 575.

[31] K U P R A D Z E , V. D . , loc. cit. p . 12.

APPENDIX

Calculation of the Green's solution

In this appendix we show how, for a homogeneous medium of

infi-nite extent, the elastodynamic Green's solution can, at least in

principle, be obtained. To this aim, we take the three-dimensional

Fourier transform of the field quantities with respect to the spatial

coordinates xi, X2 and X3. For a function / = /(«), it is defined as

Hk) ^ J j j - . . /(*) exp(-ife-*) d*. (A.l)

The Fourier inversion theorem then 3nelds

/(*) = (27t)-3 ni-oo /(*) exp(ifc • X) dk. (A.2)

A detailed discussion of the conditions under which (A.l) and (A.2)

hold can be found in [29]. Apphcation of (A.l) to the equation of

motion (cf. (2.1))

^j^Ux, *#) + p<o^uf(x, x^) = ~a{d{x - x^) (A.3)

and to the constitutive equation (cf. (2.3))

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yields

ikjff_j{k, x^) + pco^afik, x^) = —ai exp(-ife-«^) (A.5)

and

• ? u ( * . * # ) = <^i,}.P,q^kpii^(k, «a.). ( A . 6 )

EUmination of rf,j{k, x^) from (A.5) and (A.6) leads to

Ci.j,p,qkjkpU^(k, x^) — pm^iif{k, x^) = ai exp(—ife-«^). (A.7)

Introducing

a«, q = Ci.y, p, qkjkp — pOJ^t, q, ( A . 8 )

(A.7) can be rewritten as

a<.«"^(*. *#) = «i exp(—ife-*^). (A.9)

The solution of this system of hnear algebraic equations is written as

««(fc,*^) = ^ ^ e x p ( - i f e - * ^ ) , (A.IO)

in which

Zl = det(aj,y) and zlg;j = minor(a(,j). (A.l I)

With the use of the Fourier inversion theorem (A.2), the Green's

tensor Ug.^(x, x^) is then obtained as (cf. (5.4))

<;.-(*.*^) = (2«)-3J[

—^ exp[ife • {x - Xg,)] dk. (A. 12)

This shows that M^;i(*, x^) depends on Xi and x^;i only through the

difference Xi — x^-^u as would be expected in a homogeneous

me-dium. The stress tensor Tfj[x, x^) can subsequently be obtained

with the aid of the constitutive equation (A.4), or, alternatively by

inversion of (A.6), in which iJ^ is given by (A.IO). For extensive

applications of the technique outlined in this appendix, we refer to

a paper by Eason, Fulton and Sneddon [30].

In general, the evaluation of the inversion integral is not a simple

procedure. In the case of an isotropic medium, the inversion can be

carried out explicitly. Then an^q is given by

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From (A. 13) we obtain

Aq;i = {jukjkj — pa)2)[((A -f 2fJ,) kmkm — pco^)) dg,i +

-(k + fl)k(kg] (A. 14)

and

A = {nk]kj - /30)2)2 {(A -f 2n) kiki - pw2}. (A. 15)

Introducing the wave numbers

Ap = a)[/,/(A-f 2/.)]*, (A. 16)

ks = co{pl,i)i, (A. 17)

we can write the expression for Aq-^ijA as

A fi{kiki — A|) \^ kiki — kp kiki — k\)

Now, it can be shown that

{2-K) - 3

]\\

[klkl - kl s) 47ti?

dk= - ^ v - v . . . . v ( ^ j 9 ^

exp[ife-(* — «a.)] j ^ _ exp(iAp,si^)

J J J —oo

in which

i? = {{Xi - X^,i)(Xi — X^,i)}i > 0.

Using (A. 19), together with the rule that the factor ikj in the

k-domain corresponds to dj in the ^-k-domain, we finally obtain

o , 2\-i ^^ ( exp(iApi?) exp(iAsi?) \

+

Subsequently, we calculate the stress-tensor with the aid of the

constitutive equation. The result is

^.^,•:*(*. **) = ^^i.jSp^p-.k + M^i»?;k + Sjuf.k)' (A.21)

in which uf.,^ is given by (A.20). Equation (A.20) has also been given

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FAR-FIELD RADIATION CHARACTERISTICS OF ELASTIC WAVES AND THE ELASTODYNAMIC

RADIATION CONDITION ^

T.H. TAN

Department of Electrical Engineering, Division of Electromagnetic Research,

Delft University of Technology, Delft, The Netherlands

Abstract

In the present paper, a radiation condition for time-harmonic elastic waves in a homogeneous, isotropic, perfectly elastic medium is presented. It is based upon certain physical considerations pertaining to the

elastodynamic radiation generated by a source of finite extent. In it, the particle displacement on and the traction across a spherical surface around the source occur. Separate conditions are imposed on the radial and the angular parts of these quantities. It is shown that the resulting far-field radiation characteristics for the particle displacement are in accordance with the properties that could be expected from the physical considerations we have started with.

The research reported in this paper has been supported by the

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1. Introduction

One of the aspects of theoretical as well as experimental importance in the problem of radiation and diffraction of waves is their behaviour at "infinity", i.e. at large distances from the source that generates the waves. From a mathematical point of view, this behaviour is needed in connection with the uniqueness of the solution of the radiation or diffraction problem under consideration. As regards experiments, the radiated or diffracted field is often observed at relatively large distances from the source. Sommerfeld [1,2] discussed the problem for scalar waves and argued that at infinity the solution should consist of diverging spherical waves. Along the same lines. Silver [3] and Miiller [4] have formulated the corresponding "radiation condition" pertaining to electromagnetic waves. All these formulations are con-fined to waves that vary sinusoidally in time. An alternative way of determining the asymptotic behaviour of the wave motion has been pre-sented by Wilcox [5] and Stoker [6]. They have shown that the radiation condition can be obtained from a properly formulated initial-boundary value problem.

In the present paper, a radiation condition for time-harmonic elastic waves in a homogeneous, isotropic, perfectly elastic medium is presented. It is based upon certain physical considerations pertaining to the elastodynamic radiation generated by a source of finite extent. The source strength is taken to vary sinusoidally in time with circular frequency u. The complex representation of the field

quantities is used. In the formulas, the complex time factor exp(-iijJt) , in which i is the imaginary unit and t denotes the time, is suppressed. To locate a point in space, we use the Cartesian coordinates x,, x-and X, which in this order form a right-hx-anded system. Throughout the calculations, latin subscripts are to be assigned the values 1, 2 and 3; for repeated subscripts, the summation convention holds. Occasion-ally, bold-face type letters will be used to denote vectorial quanti-ties. The operator 3. denotes differentiation with respect to x..

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2. Integral representation for the particle displacement

The configuration under investigation is shown in Fig.1. In the un-bounded domain V , outside a un-bounded, sufficiently regular, closed

surface S, a homogeneous, isotropic,perfectly elastic medium is present. In a bounded sub-domain D of V , a source of primary elastodynamic radiation is present. Its operation is described through a prescribed distribution of body force with volume density f. = f.(x). Further, as in elastodynamic diffraction problems, the domain inside S can be regarded as a scattering "obstacle", being as such a source of, secon-dary, radiation. The particle displacement u. and the stress T. .

1 i,j \ V ^ h omog en cous, \ isotropic, . pc r fectty ^ elastic mc d i u m \ I I

Fig.l. Configuration for which an integral representation for the particle displacement is obtained.

associated with the elastodynamic wave motion satisfy the equation of motion [7]

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3.T. . + pu^u. = -f. (2.1)

and the constitutive equation [8]

T. . = X5. .(a u ) + u(9.u. + 3.U.). (2.2) i,J i,J P P 1 J J 1

Since the radiation condition we want to obtain is interrelated with a source-type integral representation for the particle displacement, we outline a few steps of the process of obtaining such a representation. For full details, we refer to an earlier paper [9].

We apply the elastodynamic reciprocity relation (see [9], eq. (4.5)) to the domain V , bounded internally by S and externally by S.. One of the elastodynamic states in the reciprocity relation is taken to be the field {u., T. . } , satisfying (2.1) and (2.2). The other is taken to be

1 ^»J G G

a so-called Green's elastodynamic state {u. , x. . }. The latter satis-1 i.J

fies (2.1) and (2,2), but with f. = a.5(x - x - ) , where a. is a non-zero, but otherwise arbitrary, constant vector and 6(x - x ) is the

three-di-— three-di-— r mensional unit pulse. The reciprocity relation then leads to

//„ (u.T. . - u . T. .)n. d A = //„(u.T. . - u. T. .)n. dA JJS.' 1 i,J 1 i,j' J •'•'S' 1 i,j 1 i,j J

^ (2.3) + ///ofi^i^dV - a.u.(Xp) when Xp £ V ,

In (2,3), the unit normals n are chosen as indicated in Fig.l. The integral representation for the particle displacement u. itself follows

^ G from (2,3) by taking into account the linear dependence of T . . and

G . •'"'"' u. on a. (see Section 4 ) .

3. The elastodynamic radiation condition (physical considerations) On physical grounds, the particle displacement u.(jc ) in (2.3) should be uniquely determined by the distribution of body force in D and the

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influence of the obstacle inside S, This implies that the right-hand side of (2.3) should be independent of A {aondition A).

Further, it can be argued that the acoustic power radiated through any closed surface completely surrounding D and S should be finite and positive. To this aim we impose,on S ,the stronger condition that the acoustic power flow density should be radial, positive and proportional

-2 . .

to A {oondit-ion S ) . Condition B implies that the amplitudes of both the particle displacement and the stress are inversely proportional to the distance from the source.

As in the case of scalar acoustic and of electromagnetic waves, we further expect that the interrelation between the particle displacement and the stress at large distances from the sources of radiation is locally the same as that for a uniform plane wave that travels•radially away from the source {condition C),

Further, in a homogeneous, isotropic, perfectly elastic solid there are two kinds of uniform plane waves, viz, the longitudinal or P-waves, travelling at speed c = [(X + 2y)/p] and the transverse or S-waves

" 1

travelling at speed c = (y/p) , In this respect we observe that the Green elastodynamic state for a homogeneous, isotropic, perfectly elastic solid has a particle displacement which is the sum of a curl--free part (the P-wave part) and a divergencecurl--free part (the S-wave part). The corresponding stress also separates into a P-wave part and an S-wave part (see Appendix, eq. (A.6) and (A.7)). Using these results in (2.3),we conclude that u can be written as

P S

u = u + u , (3.1)

where

curl u^ = 0^ and div u = 0. (3.2)

Application of the constitutive equation i'2.2) leads to the result that T. . can also be written as

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T. . = T. .^ + T. .^. (3.3)

i,J i.J i.J

P P

The P-wave motion is defined by {u , T. . } and the S-wave motion by s s . •'"'•'

{u , T. . }. Let the corresponding plane waves be k 1, J

^ \ ''' ^i,/''> = V ' ' '

h,"'"'

exp(ikp^3.x), (3,4)

in which

^.s'^P,s=^^/<=P,s)'- (3-5)

Substitution of (3,4) in the constitutive equation leads to

\./'%,S;j - -S,S^''' = °- (3-^)

In (3.6), k^ denotes the unit vector along k^ . Further, Z = pc and Z = pc_ denote the P- and the S-wave impedance, respectively. The wave impedances yield the ratio between the traction of a plane wave at a plane normal to the direction of propagation to the opposite of its particle velocity. In view of these properties we expect the particle displace-ment u at infinity to be of the general form

„ exp(ikp|xp|) g ^ exp(ikglxpl) u(Xp) -v A" (Xp) ^ — + A"(xp)

^HZEpI ^^lipl (3.7)

as |_Xp| -*• ">,

« . P where x denotes the unit vector along x . In (3,7), A points along

—r —T

Xp, i,e,

.P ,.P.

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S

while A is perpendicular to x^^, i,e.

A ^ - ^ = 0, (3,9)

As in the scalar and in the electromagnetic case we expect the following relation to hold

^P;j\'''

-^

i'^P,S%j\^''(V ^« l^pl

•-

-• (3-'°)

Employing the constitutive relation (2,2) and eq, (3,10), the local relation between the stress and the particle displacement at infinity is then obtained as

The order symbol in (3,11) takes into account that each term on the left-hand side is already of order 0(|x | ) . When (3.7) dnd (3.11) are

G G

used in the left-hand side of (2,3) and T. . and u. are taken to

i.J 1

consist of waves expanding from the source-point P, we indeed have

//^ (u.T. .^ - u.^T. .)n. dA = 0(A~') as A -.- ». (3,12)

Since, further, the right-hand side of (2,3) is independent of A, the left-hand side of (3,12) vanishes identically. Consequently,

a.u.(x„) = //^(u.T. .^ - u.^T. .)n. dA + //Lf.u.^dV

1 1 — P •'•'S 1 1 , J I i , J J •'•'•'Dii

(3,13) when x„ e V

-P

In the next section, the mathematical formulation of the physical considerations given in this section will be investigated,

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4. The mathematical formulation of the elastodynamic radiation condition To our opinion, the most elegant formulation of the elastodynamic ra-diation condition is

//g |2i(T.^.x. - ioiZpUjI dA = o(l) as A ->• " (A,l)

and

//„ | { T . . X . - (x.T. ,x,)x.} - ia)Z„{u. - (u.x.)x.}l dA = o(l) •^^S^' i,j J J j,k k 1 S i J J 1 '

as A ->• <». (4.2)

The formulations (4.1) and (4.2) show a certain resemblence with the corresponding condition to be imposed on the pressure of scalar acoustic waves, as introduced by Rellich [10] and the one for electro-magnetic waves,as introduced by Silver [3] and Miiller [ 4 ] , We shall prove that (4,1) and (4.2) meet all requirements that have been touched upon in the preceding section,

First of all we show that (4,1) and (4.2) imply

//.

IL

//c //c x.u.I dA = 0(1) as A x.T. .X. dA = 0(1) 1 i.J J u. - (u.x.)x.I dA = 0(1) as A ->• <», as A ->•<",

' i j ^ j ~ ^^j^j.kV^i' '^^ = °^')

as A (4.3) (4.4) (4.5) (4.6)

To this aim we rewrite (4.1) as

//„ {Z„ Ix.T. .x.l^ + u)''Z„|x.u. I ^ + ia)[(x.u. ) ( X . T . . X . ) •^^S. P ' 1 i,j j ' P' 1 i' ' - 1 1 1 i,J J

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and (4,2) as

\ ^ ' s ~ ' l ^ i , j ^ j - (^j^j,kV^il^ " "%l"i - (Vj>^l^

+ ia![(T. .x.u. ) - (x.T. .x.)(u. X.) (4,8) i-,J J i J i.J 1 J J

- ( T . . x.u ) + (x.T X ) (u.x.)] } dA = o(l) as A ^ ", i>J J ^ Jj-jJ-"- J J

*

in which denotes the complex conjugate of the relevant quantity, Addition of (4.7) and (4,8) yields

— I 9 9 9 — 1 2

//„ (Z„ Ix.T. ,X. I + 0) Z^lx.U. I + Z„ I T. .X. - (x.T. , X- )X. I •'^S^ P I 1 i,j jl P' 1 i' S ' i,j J J J.kk i'

9 9 -k -k

+ 0) Z„|u. - (x.u.)x.| ) dA = ito//„ (T . . u. - T. .U. ) X . dA s' 1 J J i' ' •'•'s^' i,j 1 i,j 1 J

+ o(l) as A + <», (4,9)

With the aid of the elastodynamic reciprocity theorem, applied to * *

{u., T. .} and lu. , T. . 1, the surface integral at the rigth-hand i' i,j 1 ' i,j '

side of (4.9) can be written as

[( (T. . U. - T. .U. )X. dA = / / „ ( T . . U. - T. .U. )n. dA •^•^S i,j I i,j 1 J •'^S i,j I i,j 1 J

(4,10) - //L(f.*u. - f.u.*) dV.

•' •'•'D^ 1 1 1 1

From (4.10) it follows that the right-hand side of (4,9) is bounded and independent of A. Since, further, the left-hand side of (4,9) is the sum of non-negative contributions, we conclude that each term in the lefthand side of (4,9) is bounded as A >• «>. As a result, (4,3) -- (4.6) follow. Note that (4.10) is in accordance with Condition B of Section 2.

We next prove that (4.1) and (4.2) are sufficient for the surface integral over S in (3.12) to vanish. In this connection it is impor-tant to note that the Green solution, defined by (A.l), (A,2), (4.21)

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