Diffraction of elastic SH-surface waves

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DIFFRACTION OF ELASTIC SH-SURFACE WAVES

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DIFFRACTION OF ELASTIC SH-SURFACE WAVES

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DIFFRACTION OF ELASTIC SH-SURFACE WAVES

PROEFSCHRIFT

TER VERKRIJGING VAN DE GRAAD VAN DOCTOR

IN DE TECHNISCHE WETENSCHAPPEN AAN DE

TECHNISCHE HOGESCHOOL DELFT, OP GEZAG VAN

DE RECTOR MAGNIFICUS PROF. IR. L HUISMAN,

VOOR EEN COMMISSIE AANGEWEZEN DOOR HET

COLLEGE VAN DEKANEN TE VERDEDIGEN OP

WOENSDAG 29 JUNI 1977 TE 16.00 UUR

door

FREDERIK LAMBERTUS NEERHOFF

elektrotechnisch ingenieur

geboren te Den Heider

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DIT PROEFSCHRIFT IS GOEDGEKEURD DOOR DE PROMOTOR PROF. DR. IR. A. T. DE HOOP

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CONTENTS

7

SUMMARY 11

CHAPTER I

SCATTERING 9F SH-WAVES BY AN IRREGULARITY AT THE MASS-LOADED BOUNDARY OF A SEMI-INFINITE ELASTIC

MEDIUM* 15 1. Introduction 15

2. Formulation of the problem 16 3. Integral representations for the

scattered wave function 18 4. Integral equations 21 5. Expressions for the surface wave

amplitudes 22 6. Expressions for the far-field

radiation pattern 23 7. Discussion of the numerical

techniques 26 8. Numerical results 27 Appendix A 32 Appendix B 34 References 35 CHAPTER II

SCATTERING AND EXCITATION OF SH-SURFACE WAVES BY A PROTRUSION AT THE MASS-LOADED BOUNDARY OF AN

ELASTIC HALF-SPACE** 37 1. Introduction 37 2. Formulation of the problem 38

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8 _CONTENTS

4. Integral equations 41 5. Surface wave amplitude and

far-field radiation pattern 42 6. Method of computation 43 7. Numerical results 44

References 49 Appendix 49

CHAPTER III

DIFFRACTION OF LOVE WAVES BY A STRESS-FREE CRACK OF FINITE WIDTH IN THE PLANE INTERFACE OF A

LAYERED COMPOSITE 51 1 . Introduction 52 2. Formulation of the problem 54

3. General discussion on the fields

in the uniform structure 58 4. Integral representation for the

scattered field 60 5. Integral equation for the jump in

the particle displacement across

the crack 63 6. Method of solution of the

differential-integral equation 63 7. The scattered field in the far-field region 66

7.1 The amplitude radiation

characteristic of the radiation field 69 7.2 The amplitudes of the Love-wave

contribution 71 8. The scattering matrix of the crack 71

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CONTENTS

9

10. Numerical results

Appendix A. Incident fields

Appendix B. Green's function for the uniform, layered composite Appendix C. Reciprocity relations Appendix D. Power-flow relations References 80 99 101 104 111 113 SAMENVATTING

115

LEVENSBERICHT

120

This part has been reprinted from the Proceedings of the Royal Society of London, Series A, 1975, 342, pp. 237-257, in which it has been published.

** . . . . . .

This part has been reprinted from Applied Scientific Research, 1976, 32, pp. 269-282, in which it has been published.

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SUMMARY

As in other branches of mathematical physics, the -integral equation approach for analyzing the diffraction of elastic surface waves by dis-continuities of finite extent in a semi-infinite elastic medium or an elastic slab, appears to be a fruitful one. In this respect, we think about surface waves (Rayleigh- or Love waves) as well as discontinu-ities (surface deformations, inclusions or cracks) of a rather general kind, while the semi-infinite elastic medium or the slab may be uniform or horizontally layered. The integral equations eventually obtained for this class of diffraction problems, have at least one property in common that gives rise to serious difficulties in the numerical treatment of them. In the derivation of the integral equation, we base ourselves on the integral representation for the particle displacement in the config-uration under investigation. This integral representation follows from a suitable reciprocity relation, when in the latter a suitable Green's state is substituted for one of the elastodjmamic states occurring in it. By letting the point of observation approach the boundary of the discon-tinuity, the integral equation results. Let us now consider the Green's state in some more detail. This state pertains to a situation in which either a point source (three-dimensional problem) or a line source (two-dimensional problem) is placed in the "undisturbed" structure under

in-vestigation. The solution of the so constructed boundary-value problem is called the Green's solution or the Green's function for that particu-lar configuration. Now, in our applications the Green's solution has to be represented as a Fourier inversion integral with respect to the hor-izontal coordinate(s). Further, it can be shown that the surface wave(s) generated in the Green's state, are related to poles in the integrand of the Fourier inversion integral. If the medium is taken to be lossless, these surface-wave poles are located at the real axis of the complex plane in which the path of integration of the Fourier inversion

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-12 SUMMARY

integral is situated. As a consequence, the numerical evaluation of the Fourier type of integral (which occurs as kernel function in the inte-gral equation) has to take place in the complex plane in order to avoid numerical instabilities due to the presence of the surface-wave poles. Summarizing, we observe that the numerical evaluation of the infinite Fourier type of integral, which occurs as kernel function in the rele-vant integral equation, in the complex plane is a common difficulty to

the class of diffraction problems described here.

In the present thesis we investigate the diffraction of time-har-monic, elastic SH-surface waves by discontinuities of finite width. The problems to be considered are two-dimensional. The main tool in the anal-ysis, as mentioned above, is the integral-equation formulation of the problems. Elastic SH-surface waves are of a scalar character and, as a consequence, are the simplest possible elastic surface waves. Now, the simplest configuration in which SH-surface waves can propagate is the mass-loaded isotropic half-space which, under certain conditions, may be

regarded as a low-frequency approximation to the Love-wave configuration. A more complicated structure is the actual Love-wave configuration itself. For both configurations, we have solved a number of diffraction problems. The resulting integral equations, with their Fourier type of integral as kernel function, are solved numerically. In the diffraction problems un-der investigation, the incident wave is taken to be either a uniform plane bulk wave or a non-uniform surface wave. In each case, we compute the transmission and reflection factors, the amplitude patterns of the launched surface waves as well as the far-field radiation patterns of the scattered bulk wave.

In Chapter I, we investigate the scattering of SH-surface waves by an irregularity at the mass-loaded boundary of an elastic half-space. Two types of surface discontinuities are considered, viz. an indentation and a plane discontinuity in mass-loading. Both irregularities are of fi-nite width. For the two types of irregularities, the boundary-value

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SUMMARY 13

problem (which is of the third kind) is solved by employing the integral-equation approach. The resulting integral integral-equations in turn are solved / numerically by the application of the method of moments, in which the un-known quantity is approximated by a sequence of step functions. A reci-procity relation is derived that exists between the amplitude of the launched surface wave in the case of an incident bulk wave, and the far-field radiation pattern of the scattered bulk wave in the case of an in-cident surface wave. Numerical results are presented for the following configurations: a triangularly-prismatic indentation, a rigid strip load-ing and a traction-free interruption in the mass-loadload-ing.

In Chapter II, the diffraction of SH-surface waves by a protrusion at the mass-loaded boundary of an elastic half-space is investigated. The boundary-value problem (of the third kind) is solved by employing two suitably chosen Green's functions. One of them is taken to be the half-space Green's function and is represented as a Fourier type of integral, the other is taken to be the Bessel function of the second kind and order zero; its domain of definition is the bounded domain occupied by the pro-trusion. By employing the integral-equation approach, a system of three coupled integral equations results. This system is solved numerically by using the method of moments in which the appearing unknown quantities are approximated by a sequence of step functions. A reciprocity relation is shown to exist between the amplitude of the launched surface wave and the amplitude radiation pattern of the scattered wave in the far-field region. Numerical results are presented for a triangularly-prismatic protrusion; they are compared with the results pertaining to a corresponding indenta-tion in the mass-loaded boundary, that have been obtained in Chapter I. It turns out that for relatively small widths of the irregularities, the scattering properties of a protrusion differ from those of an indenta-tion, especially for non-shallow deformations. For larger values of the widths of the irregularities under consideration however, and especially for relatively shallow perturbations, the scattering amplitudes of the

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14 SUMMARY

two types of irregularities behave quite similar, apart from a phase shift.

In Chapter III, we treat the diffraction of Love waves by a stress-free crack of finite width in the plane interface of a layered composite. The resulting boundary-value problem for the unknown jump in the particle displacement across the crack is solved by employing an integral-equation approach. Then, an differential-integral equation results. Next, the un-known quantity is expanded in terms of a complete sequence of expansion functions in which every term separately satisfies the edge condition. This leads to an infinite system of linear, algebraic equations for the coefficients in the expansion. The elements of the matrix of coefficients of this system are Fourier type of integrals, corresponding to the Green's function of the uniform, layered structure. The relevant system is solved numerically. Furthermore, the scattering matrix of the crack, which re-lates the amplitudes of the outgoing waves to the amplitudes of the in-cident waves, is computed. Several reciprocity and power-flow relations are derived. Numerical results are presented for a range of material con-stants and geometrical parameters.

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Scattering of SH-waves by an irregularity at the mass-loaded

boundary of a semi-infinite elastic medium

B Y F . L. N E E E H O F F

Department of Electrical Engineering, Division of Electromagnetic Research, Delft University of Technology, Delft, the Netherlands {Communicated by D. S. Jones, F.R.8. - Received 30 April 1974)

A rigorous theory of the scattering of two dimensional SH-waves by an irregularity at the mass-loaded boundary of a semi-infinite elastic medium is presented. Two types of surface irregularities are considered: (a) an indentation, (6) a discontinuity in mass-loading. The incident SH-wave is taken to be either (1) a bulk wave or (2) a surface wave. For the two types of irregularities, the corresponding boxmdary value problem (of the third kind) is solved by employing a suitably chosen Green function; the latter is represented as a Fourier type of integraL This procedure leads to integral equations in which the relevant field distributions on the disturbed parts of the boundary occur as unknown quantities. In case (1) the ampli-tudes of the launched surface wave are computed; in case (2) the trans-mission and the reflexion factor are computed. For both cases, expressions are obtained for the far-field radiation pattern of the scattered bulk wave. I n an appendix a reciprocity relation is derived between the amplitude of the launched surface wave as a function of the angle of incidence in case (1) on one hand and the far-field radiation pattern of the scattered bulk wave in case (2) on the other hand. Numerical results are presented for the following configurations: a triangularly prismatic indentation, a rigid strip loading and a traction-free interruption in the mass-loading.

1. I N T B O D U C T I O N

Elastic surface waves have received considerable attention because of their geo-physical interest, whereas in the last few years there is a wide field of application in micro-electronics (for an extensive bibliography see, for example. White 1970). In both fields of application, it is of importance to investigate the scattering of elastic waves by surface irregularities. Approximate studies pertaining to P- and SV-waves have been undertaken by, for example, Gilbert & Knopoff (i960), Abubakar (1963), Levy & Deresiewicz (1967) and Mclvor (1969) and, pertaining to SH-waves by Slavin & Wolf (1970) and Aki & Lamer (1970). The subject of investigation of the present paper is the scattering of SH-waves in a semi-infinite, mass-loaded medium on which a surface irregularity is present. As pointed out by Alsop, Goodman & Ash (1971), the mass-loaded boundary is a very good low-frequency approximation for describing the Love wave configiuration. For this reason, our results are of importance in connexion with the scattering of Love

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-16 SCATTERING OF SH-WAVES

waves. We discuss the following types of surface irregularities: (a) an indentation in the mass-loaded boundary, (6) a discontinuity in mass-loading. Both types of irregularities are assumed to be cylindrical and of finite width. The surface mass-density of the applied loading at the indentation may differ from the one of t h e undisturbed mass-loading. The time-harmonic incident SH-wave is taken to be either (1) a bulk wave or (2) a surface wave. For both types of irregularities, the corresponding third boundary value problem is solved by employing a suitably chosen Green function; the latter is represented as a Fourier type of integral. The procedure leads to integral equations in which the field distributions on the disturbed parts of the boundary occur as unknown quantities. Next, the launched surface wave amplitudes in case (1) and the surface wave transmission and reflexion factor in case (2) are expressed in terms of these field distributions. Furthermore, for both cases the far-field radiation pattern of the scattered bulk wave is computed, again in terms of these field distributions. Once the resulting integral equations have been solved, we present numerical results for the following configurations: a triangularly prismatic indentation, a rigid strip loading and a traction-free inter-ruption in mass-loading. In appendix A, we present the derivation of the integral equation pertaining to a discontinuity in mass-loading in the general context in which the Green function satisfies a boundary condition of the third kind on a contour of arbitrary shape. Finally, in appendix B we derive a reciprocity relation that exists between the amplitude of the launched surface wave as a function of the angle of incidence in case (1) on the one hand and the far-field radiation pattern of the scattered bulk wave in case (2) on the other hand.

The analysis of the different surface irregularities shows minor but important differences, particularly in the integral equations to be obtained. Therefore, t h e major steps have in each case been indicated explicitly.

2. F O R M U L A T I O N O F T H E P R O B L E M

The position in space is specified by the Cartesian coordinates x^, x^ and x^, which in this order form a right-handed system. Let us consider a mass-loaded elastic medium occupying the half-space

— 00 < a;i < 00, — oo < Xj < oo, 0 < Xj < oo.

The surface mass-density of the appUed loading is constant and denoted by cr-^. Along the surface, an irregularity is present, which is assumed to be cylindrical in the ij-direction and bounded in the aij-direction. Two types of surface irregu-larities will be investigated: (a) an indentation in the mass-loaded boundary and (6) a discontinmty in mass-loading. The surface mass-density a^ of the applied loading at the indentation may vary with position and differ from Cj. The surface mass-density of the discontinuity in mass-loading, too, is denoted by cr^. At this point we remark that if CTJ -> oo the boundary of the irregularity under consideration tends to a rigid one, and if o-j-^-O the boundary tends to a traction-free one. The

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SCATTERING OF SH-WAVES 17

bulk medium is assumed to be linear, isotropic, homogeneous and perfectly elastic; its mechanical properties are fully characterized by the Lame-constants A and /t and the volume mass-density p (figure 1).

The time harmonic incident SH-wave is taken to be propagating in the Xj^, x^-plane. The complex representation of field quantities is used; the complex time factor exp (-iw<) is omitted throughout. Furthermore, the summation convention will be used; the repeated Greek subscripts a and /? are to be assigned the values

PioUBE 1. Surface irregularities and incident waves.

1 and 3. The «j-component of the particle displacement (which is the only non-vanishing component) is denoted by U. In the bulk medium we write

t/(«) = t7»(«) + C7»(«), (2.1) in which x = («i.Xj) denotes the two dimensional position vector, and where f7'

is defined by

Tji _ I ^ - w . + ^ . w . ''^len a bulk wave is incident, 1 .^^, ~ Iü^ari. when a surface wave is incident.)

In (2.2), t7|,.w. denotes the incident bulk wave, ül_y,, is the reflected bulk wave arising from reflexion against the plane boundary of the semi-infinite medium in the absence of any irregularity; U\%_ denotes the incident surface wave, travelling in the positive or the negative Xj-direction respectively (figure 1). The wave function f7' occurring in (2.1) is denoted as the scattered field, i.e. the deviation of U from ?7' due to the presence of the irregularity under consideration. In its domain of definition each of the wave functions satisfies the source-free elastodynamic wave equation for SH-waves

(K^a + ^s){V,UKV-} = Q, (2.3)

where 6^ denotes the partial derivative with respect to x^, and fcg = w/cg denotes the wave number of shear waves. (Here, Cg = (/f//^)* denotes the phase velocity of shear waves.) At those points where the properties of the medium change abruptly, the differential equations (2.3) have to be supplemented by boundary conditions. Let re, denote the unit vector in the direction of the normal to the boundary as shown in figure 1. Then it follows that T^^^^n^ = -crui^U upon approaching a mass-loaded boundary. (Here, 7J_j denotes the complex stress tensor, in which the sub-scripts i and j are to be assigned the values 1,2 and 3.) On account of the constitutive

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18 SCATTERING OF SH-WAVES

relation for SH-waves, namely T^j =/*6^Z7, the boundary condition can be re-written as

(re, S, 4- Agi/) (7 = 0 at a mass-loaded boundary, (2.4) in which rj denotes the reduced specific acoustic impedance of the mass-loading and

is defined by

V = ^sf^/P- (2.«)

I t is noted that tj = rj^ii the surface mass-density of the mass-loading is (Ty and

ij = Tj^ii this quantity equals cr^. From (2.5) it follows that (2.4) represents the

condition at a rigid boundary in the limit 7;->oo(C/ = 0 a t a rigid boundary) and the condition at a traction-free one in the limit i/ ->• 0 (re, 6, J7 = 0 at a traction-free boundary). Let the incident waves I/j,^. and C/^*, be plane waves of unit amplitude and zero phase at the origin. Then

Ui.y,.(x) = exp (io^xi-irlxj), (2.6)

in which a4 = igsin(ö') and y | = fcsCos(ö'), (2.7) where ö'( —JTT < Ö' < \-K) denotes the angle of incidence as shown in figure 1. In accordance with (2.4), with T; = rjy, the incident surface wave is given by

C^8:^.(*) = exp ( ± ik^xy - ksViXa), (2.8) in which k^ = k^ii+Vi)^ > 0. (2.9)

In (2.8), exp( + ik^rj) denotes the propagation factor of the surface wave in the ajj-direction and exp (— kgrj-^x^) denotes its exponential decay factor in the positive aTs-direction. From the definition of the reflected bulk wave JT^.w. and the boundary

condition (2.4) it follows that

Ul,„,(x) = R^exp{ia^xy + iy^x^), (2.10)

in which R^ denotes the reflexion factor of {7},.^. against the undeformed mass-loaded boundary; it is obtained as

Ri = (iy^ - ksViWvh + *s^i)- (2.11)

Furthermore, the wave function U" should satisfy the radiation condition per-taining to (2.3) and (2.4). This implies that at large distances from the disturbance

U' should consist of cylindrical bulk waves and plane surface waves, both travelling away from the irregularity under consideration.

3. I N T E G R A L R E P R E S E N T A T I O N S F O R T H E S C A T T E R E D W A V E F U N C T I O N

In this section we derive integral representations for the scattered wave fimction Ï7'. In the derivation of these representations we employ a suitably chosen Green function, to be denoted by O. I t has to satisfy the inhomogeneous two dimensional Helmholtz equation

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in which «^ denotes the position vector of a point 0* and S denotes the two dimen-sional Dirac delta distribution. We construct 6 in such a way that

lim (63 -)- fcg7/i) G(x; x^) = Q if - 00 < Xj < 00, (3.2)

ir,-»0

with «1 # a;i_^ as Xj^^^-O. Furthermore, Q should satisfy the radiation condition. The solution of (3.1) subject to the boundary condition (3.2) can be represented as a Fourier inversion integral with respect to the spatial coordinate x^. The result is

G{x;Xf) = (i/47r)|^7jHexp(iysN3-a;3.^1)

-I- iJg exp {iys(a;3 -I- ^a.^.)}] exp {i*(a;i - Xy_^)) dfc, (3.3) in which yg = ys(i) is defined as

7s = ( 4 - ^ ¥ , (3.4)

with Re (yg) g 0 in the entire complex i-plane, and ifg = R^{k) as

•Kg = (irs-^s7i)/(irs + *si7i). (3.6) The path of integration J" in (3.3) is situated in those parts of the complex

A-plane where both Re(yg) > 0 and Im(yg) > 0. From (3.4) and (3.5) we observe that the singularities in the integrand are: a branch point at fc = + ifcg and a simple pole at ifc = + fcm. The latter arises from the simple zero of the denominator in the expression (3.5) for iJg = Rt^{k) and represents the surface waves travelling away from the source point 0*.

A principal tool in the derivation of our integral representation for TJ^ is the two dimensional form of Green's second identity

j j (9'9a9<,'»-M8a9a9')da;ida;3 = i {gvji^u-uvj>^g)di8, (3.6) in which v„ denotes the unit vector in the direction of the outward normal to the

boundary '€ of the domain 2 and s denotes the arc-length along the boundary. Starting with the indentation, we apply (3.6) to the domain Dj, whose boundary is chosen to consist of the indentation L, the undeformed part of the boimdary of the mass-loaded half-space C^, (union of the segments Xj = 0, ~h <Xy< —a and 0:3 = 0, a < a;i < h), the vertical straight fines C|ert (^1 = 6,0 < a;, < 6) and C^ert (aij = — 6,0 < «s < 6) and the horizontal straight line Ci,„ («3 = 6, ~h < x^K h) as shown in figure 2.

In (3.6) we employ the functions {g, u) = {0, Ü'}, the point of observation ^ being an ulterior point of D^. With reference to (3.6), we now consider the contri-butions of Cm + C^irt + Ciior + ^TCrt to the integral in the limit 6->oo. The integral along Cm vanishes on account of the boundary conditions (2.4) and (3.2) on C„ for U^ and O respectively. Moreover, it can be shown that the integrals along C|=ert and Cjjor also vanish in the limit 6->.oo on accoimt of the radiation condition imposed upon both U' and G. Hence, on account of (2.3) and (3.1) ,we arrive at

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in which w, denotes the unit vector on L as shown in figure 2. The next step consists of applying (3.6) to the domain D^, bounded by the straight line

'S {X3 = 0, —a < Xi< a)

and by L as shown in figure 2. This time we employ in (3.6) the functions

{g,u} = {G,U%

while ^ is still an interior point of D^. On account of the boundary conditions (2.4) and (3.2) on S for C7' and G respectively, the integral along S vanishes. Then, on account of (2.3) and (3.1) we obtain from (3.6)

0 = f {t/'(«)»,^aG{s;x^)-G(s;Xg,)re,3, &'(«)}d«, (8.8)

x^a Cm i , = 6

in which re, again denotes the unit vector on L as shown in figure 2. Next, we write i7» = C7 — C/' in the integrand of (3.7) and subsequently add the result to (3.8). On account of the boundary condition (2.4) on L for V, we obtain

(3.9)

(3.10)

V\Xf) = J ^ V{s) (n, 9, + fcg9?j) G(s; x^) d«

for a non-rigid boundary or, by using (7 = 0 at a rigid boundary,

U'{x^) = - f {re,a,Z7(5)}(?(«;*^)ds

for a rigid boimdary. The expressions (3.9) and (3.10) are the desired integral representations for U^ in the case of an indentation. In order to obtain an integral representation for U^ in the case of a discontinuity in mass-loading, most of the results obtained above can be reused. To this aim we consider the geometry of the discontinuity in mass-loading as the limiting case of the geometry of the indenta-tion when its overall depth tends to zero. As a consequence, replacing L by S (figure 1), M,9, by 83 and by using d^G = — ^gi/iö on S, (3.9) and (3.10) reduce to

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for a non-rigid discontinuity or

C^'(**) = -jjS3Ü(s)}G{s;x^)ds, (3.12) for a rigid strip loading. The expressions (3.11) and (3.12) are the desired integral

representations for U' in the case of a discontinuity in mass-loading. From (3.9), (3.10), (3.11) and (3.12) the particle displacement everywhere in the bulk material can be calculated, provided that for the indentation the field distributions U or re,9,C/ on L and for the discontinuity in mass-loading Ü or d^U onH are known In the next section we shall derive integral equations for these unknown quantities.

4. I N T E G R A L E Q U A T I O N S

In order to obtain integral equations for the as yet unknown field distributions we determine the limiting values of the expressions (3.9) and (3.10) or (3.11) and (3.12) when the point of observation 0 approaches L or 2 respectively. Again starting with the indentation we first take the limit when 0 approaches L in (3.7). This jdelds

WM = jjü'(s)n,d,G{s;s^)-G(s;s^)nJ,U'(s)}ds if s^eL, (4.1)

in which f denotes the Cauchy principal value of the relevant integral. Next, we take the Umit when 0 approaches L in (3.8). We obtain

-iU^M=j-JÜ'(s)n,d,G{s;s^)-G(s;s^)n,d,U\s)}ds if s^eL. (4.2)

Addition of (4.1) and (4.2) and using (2.1), together with the boundary condition (2.4) on L for U, yields

iU{s^)-ü'M=.j-^U{s){n,d, + ksVz)G(s;s^)d8 if a ^ e L , (4.3) for a non-rigid boundary or, by using Z7 = 0 at a rigid boundary,

W(sg.) = i{n,d,ü(s)}G{s;s^)ds if s^eL, (4.4)

for a rigid boundary. Equations (4.3) and (4.4) are the desired integral equations pertaining to an indentation; from them U or n^S^U on L can be calculated. In ordertoobtainthelimiting values of (3.11) and (3.12) in the case of a discontinuity in mass-loading, we start with the general result obtained in appendix A. Hence

U^s^) = -j^G{8;s^)(d3 + ksVi)U^{s)ds if a ^ e S . (4.5)

Writing U' = f7— {/' in (4.5), we obtain on account of the boundary condition (2.4) on S for both fJ' (i.e. 7/ = i},) and U (i.e. r/ — i;,)

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for a non-rigid discontinuity, or from (4.4)

U\s^)=j-Jd^U(s)}Gis;8^)ds if s^e-L, (4.7)

for a rigid strip loading. Equations (4.6) and (4.7) are the desired integral equations pertaining to a discontinuity in mass-loading; from them Ü or d^U on 2 can be calculated.

5. E X P R E S S I O N S F O B T H E S U R F A C E W A V E A M P L I T U D E S

In this section we shall derive expressions for the complex amplitude .4± of the launched surface wave in the case when a bulk wave is incident upon the irregularity under consideration. (Here, the plus and the minus sign refer to surface waves travelling in the positive or the negative Xj-direction respectively.) At the same time, however, expressions will be obtained for the transmission factor T and the reflexion factor R in the case of an incident surface wave. For both cases we have to evaluate the ampUtude of the outgoing surface wave contribution to U'. If we denote this amphtude by fi±, it foUows that

A± = B±, (6.1)

when a bulk wave is incident, whereas T and R are defined by

T=l+B+ and R = B', (5.2)

when a surface wave travelling in the positive Xi-direction is incident. As a principal tool in the evaluation of the ampUtudes B+ and B~ we employ a reciprocity relation that directly follows from Green's second identity (3.6). Let W^ and Wg denote two possible wave functions, occurring in two different situations. Let they satisfy the source-free elastodjoiamic wave equation in some domain 2 situated in the a;i,a'3-plane. Furthermore, let "^denote a simply closed contour in 2. Then, em-plojdng the functions {g, v) = {ï'^, W^ in (3.6), we obtain

{'PAK^J'B~'PBV.K'PA)AS = 0. (6.3)

Starting with the indentation, we first apply (5.3) to the domain Dj, bounded by the closed contour L-I-0^,-f C^ert-t-Cupj-t-Cyert as shown in figure 2. In (5.3), the wave function with subscript A is chosen to be the scattered field in the actual situation; hence, IP^ = V^. The wave function with subscript B is taken to be an auxiliary field W^ = U^^. As we have done in §3, we shall first consider the contributions from 0 ^ + C ^ H + Cuor + Cyert in the hmit 6-»oo (see figure 2). The integral along 0 ^ vanishes on accoimt of the boundary condition (2.4) on C„ appl3dng to both ï'^ and W^. Furthermore, it can be shown that only the integral along C|=ert arising from terms containing surface waves travelling in opposite directions yield a non-vanishing contribution. After performing in the Umit 6->oo the integration along C^ert» ^ ^ arrive at

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As a next step, we apply (5.3) to the domain Dj, bounded by the closed contour 2 - F L as shown in figure 2. This time the wave function Wj^ is chosen to be the total incident field in the actual situation, i.e. W^ = Ï7', while the auxiliary field is

Wg = Ü7i;^.. Then, on account of the boundary condition (2.4) on 2 applying to

both IPj and y g , the integral along 2 vanishes. Hence, we obtain from (5.3)

I

(C7ire,8, f7'±,.- f/i±^.re,9, ü^)ds = 0. (6.S) Next, we write J7^ = C/ — C/' in the integrand of (5.4) and subsequently add the

result to (5.5). Upon using the boundary condition (2.4) for i7 on L and substituting the expression (2.8) for f j ^ in the result, we obtain

B± = (iksriylk^)\u(s){n^d^ + k^ri^)e:s.^( + ik^Xy-ksriiX3)ds, (5.6)

for a non-rigid boundary or

B± = -(iksriyih^)\ {n^\U(s)}e,x^(+ik^Xy-h^riiXi)ds, (5.7)

for a rigid boundary. Equations (5.6) and (5.7) constitute the desired expressions for JB± in terms of t/ or re, 9, Z7 on L in the case of an indentation. In order to ob-tain an expression for 5 * in the case of a discontinuity in mass-loading, we again consider the geometry of the relevant configuration as the limiting case of an indentation. As a consequence, replacing L by 2 , re, 9, by 83 and by using

(5.6) and (5.7) reduce to

B±=(iksVilK)h\{Vt-Vi)V(s)e:iLp( + ik^Xj)d8, (5.8)

for a non-rigid discontinuity or

B± = -(iA;s%/i„)J^{93?7(«)}exp(+ifcma:i)ds, (5.9)

for a rigid strip loading. Equations (5.8) and (5.9) constitute the desired expressions for £± in terms of t/ or 93 f7 on 2 in the case of a discontinuity in mass-loading. By using (5.1) and (5.2) the launched surface wave amplitude A± in the case of an incident bulk wave as well as the transmission factor T and the reflexion factor R in the case of an incident surface wave can be calculated.

6. E X P R E S S I O N S F O R T H E F A R - F I E L D R A D I A T I O N P A T T E R N

In the present section expressions wfil be derived for the far-field radiation pattern P of the scattered bulk wave. This pattern P = P(^) is introduced by writing the non-surface wave part of the scattered field, to be denoted by U%, as

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in which r denotes the distance from the origin to the point of observation 0 and ^ denotes the angle that r makes with the positive ajj-axis such that

x^^= — rsin(0),2;3_^ = rcos({5)

with — JTU < 0 < JTT. In order to obtain an expression for P = P(^), we need an asymptotic expansion for the scattered field Z7' at large distances from t h e origin. To that aim we start with the integral representations (3.9) or (3.10) and (3.11) or (3.12) for f/=, depending on the type of irregularity we are dealing with. Next, we consider the representation for the Green function G and try to obtain an expansion of G as r->oo, — Jrc < ^ < JTC. To this aim we write

(?(*; Xf) = G^[x; x^) + V(x; x^), (8.2) in which (?Q and V represent the terms of G associated with the primary line source

and its reflexion in the plane X3 = 0, respectively. From (3.3) we obtain

ö„(*;*^) = (ii)ff,S«(isPi). (6.3) where H^^ denotes the Hankel function of the first kind and order zero and

Pi = {{Xi-Xi_^y + {x3-X3_^)^)i > 0,

and F(a;;a;^) = (i/47c) ys^iJsexp{iyg(x3-l-X3_^)-t-iA;(a;i-Xi_^)}dA. (6.4) Since p^ ~ r4-a;isin(^)-a;3cos(0) as r->oo we obtain, using the asymptotic ex-pansion of the Hankel function for large arguments

(?„(*;«,) =: (ii)(2/7tA;sr)iexp(ii:gr-ii7r)

X exp {ifcg sin (^)Xi-ifcg cos (0)^:3} as r->oo, - JTC < $5 < JTT. (6.5) In order to arrive at an asymptotic expansion for F, we first substitute in the integrand of (6.4)

Xi-Xi_^ = p^amid)

and Xa+x^^^ = p^coaid) with —Jn < Ö < JTC

a n d Pi = ({«1 - 2=1.^)' -I- (2:3 + ^3.#)')* > 0.

As a next step, we deform the original path of integration F into the path

k(t) = kgsia(d+it)(~<x> <t < 00),

which is a branch of a hyperbola. The supplementing arcs C„ at infinity yield a vanishing contribution (figure 3).

Next, we apply the method of stationary phase (see, for example, Jones 1964) to the resulting integral in order to obtain its asymptotic expansion as r->Qo, which impUes p^-^00. The point of stationary phase is t = 0. Since

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as r -> 00, we finaUy arrive at

V{x;Xf) ~ (ii) (2/7i:fcgr)iexp (ifcgf- JiTt)i?,

xexp{ifcgsin(^)a;iH-iigCos(^)a;3} as r ^ o o , —^TC < ^ < ^TC, (6.6) in which R^ = iJo(^) is given by

Ro(^) = (ifcg cos (4>) - ksrij)l{iks cos (^) + fcgi/j). (8.7)

Substitution of (6.5) and (6.6) in (3.9), (3.10), (3.11) and (3.12) yields

A-plane

(6.8)

(6.9) FiouBE 3. Deformation of the path of integration.

P(«5) = f Uis)(n,Q^ + ksVi)F{Xi,X3;^)ds, for a non-rigid indentation or

P(<f>) = -j^ K 9. ü{8)} F(x„ x^; ,f>) d8,

for a rigid indentation, where F{Xi, £3; ^) is given by

F{Xi, «s; ^) = exp (ifcg sin ((p) x^ — ifcg cos (0) x^} +

Rgi^) exp {ifcg SLQ ((l>)xi + ifcg cos (^) «3} (6.10)

and P(yi) = fcgf (72-5/i)C/(«)P(«i,0;?i)d5 (6.11) for a non-rigid discontinuity or

P{<^) = -J^{93Ï7(s)}P(Xi,0;?5)d«, (6.12)

for a rigid strip loading. I t is noted that the relations (6.8) up to and including (6.12) hold for an incident bulk wave as weU as for an incident surface wave.

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7. D I S C U S S I O N O F T H E N U M E R I C A L T E C H N I Q U E S

The main difficulty in the numerical solution of the integral equations derived in §4 lies in the numerical evaluation of the Green function G which occurs as kernel function. Let us consider the integral representation (3.3) for G in some more detail. As pointed out before, G consists of two terms corresponding to the primary source and its reflexion in the plane x^ = 0, respectively. As the primary source term is nothing but a Hankel function, it can be computed by a fast standard routine for the Hankel function. In order to get a tractable expression for the re-maining term in (3.3) (cf. (6.4)), we substract a Hankel function from it and write

G(x;x^) = (ii){//,S"(fcg \x-x^\)+H§-Kks\x-Xj,\)} + I(x;x^) (7.1)

in which x^ — {x^^, —x^^^). The remaining integral I(x;x^) is then uniformly convergent. Since yg is an even function of fc, the integral can be rewritten as

7(*;*^) = ( - i f c g W - ) f ^^P{'ys(^3+^3,^)}oosWa^i-^i.^)}d^, (7 2) s/i/ yj^^ ys(iys+^s';i)

The path of integration P+ that is selected for the numerical evaluation of the integral in (7.2) is shown in figure 4.

Im(A)

^////////y/ii/ii/i////////// ni,„i,„iiiiiiin„^i,ii,„ii. Re (ft)

A-plane

r*

FiQUKE 4. Selected path of integration.

The numerical evaluation of (7.2) is performed by employing a complex trape-zoidal integration rule along P+ up to fc = fcj (figure 4). At the part of P+ along the real axis the integrand is purely imaginary and therefore a real integration procedure is used which reduces the computation time by a factor of two. From fc„ onwards (figure 4), an asymptotic approximation of the integral is used. The choice of fcj (figure 4) and k^ is determined by trial and error attempting to arrive at a minimum in the computation time. In this respect the location of the pole at fc = fc^^ is of special importance. Now that the evaluation of the kernel function has been dis-cussed, we return to the problem of solving the integral equations (4.3), (4.4), (4.6) and (4.7). This is performed by the appUcation of the method of moments (Harrington, 1968). We shaU Ulustrate the method by considering the integral equation (4.3). The unknown function, i.e. TJ on L, is approximated by M step functions,whilethecontour Lis subdivided into I f intervals L^(m = 1,2,3,..., J / ) . Subsequently, the point & is chosen in the centre of each interval L„. Then, the

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integral equation is replaced by an approximate system oi Mx M linear algebraic equations, the coefficients of which are formed by the integrals of the kernel function on the interval L„ (re = 1,2,3, ...,M). If s ^ e L „ , the kernel function is decomposed into a singular and a regular part, according to (7.1). The integrals of the regular parts can be evaluated by a numerical integration rule; this procedure also applies if s^ ^ L„, i.e. when the kernel function is regular on L„. The convergent improper integrals of the singular parts of the kernel function are computed ana-lyticaUy replacing each L„ by a straight arc. Finally, with the aid of a standard numerical inversion procedure, the unknown distribution of f7 on L is obtained. In exactly the same way, the solution of the integral equations (4.4), (4.6) and (4.7) is obtained.

8. N U M E R I C A L R E S U L T S

In the present section we perform the computations expUcitly for a triangularly prismatic indentation, for a rigid strip loading and for a traction-free interruption in the mass-loading with i/^ = 0.27r. This value of r/^ may be used e.g. in a configura-tion consisting of a semi-infinite berylUum substrate with a layer of tungsten on it, as used by Tournois & Lardat (1969) in their experiments with Love wave delay lines. Taking a layer of thickness 0.01 times the wavelength of the bulk wave in the beryUium substrate, we obtain rji ~ 0.27t. The relative error we make in applying the mass-loading description in determining the wave number of the surface wave instead of the exact boundary conditions for the layered structure is about 3 % . For the mentioned value of T/^ we have fcm/fcg — 1.18 and A„,/Ag ~ 0.85, in which A„, denotes the wavelength of the surface wave (A^ = 27c/fcm) and Ag denotes the wavelength of the bulk wave (Ag = 27t/fcg). The depth of the indentation is taken in the order of the depth of penetration d of the surface wave into the semi-infinite elastic medium. The latter quantity follows from (cf. (2.8)) rf/Ag = (27c7/i)~i; with

111 = 0.2ii: we obtain d/Ag ~ 0.25.

In figure 5 and figure 6 the amplitude J4+ of the launched surface wave is plotted as a function of the angle of incidence Ö' of the bulk wave for different values of the width of the irregularities. I t is noted that the irregularities under consideration are symmetrical with respect to the plane x^ = 0, hence, A~(d') = A+( — 6^). From figure 5 it is observed, as is to be expected, that in the limit of zero width of the indentation, keeping the depth fixed, \A±\ tends to zero at Ö' = 0. I t is also observed that the largest value of |^±(ö')| for a given width 2o of the triangular indentation shows a maximum when 2a is nearly equal to A^,. The maximum is more pronounced for a larger depth of the indentation; no such a maximum occurs for a rigid strip loading or for a traction-free interruption in the mass-loading. The results pre-sented in figure 5 and 6 are among others important in connexion with the excita-tion of surface waves in elastic materials. Moreover, using the reciprocity relaexcita-tion derived in appendix B, the conversion from surface-to-bulk waves also follows from these figures. This conversion can be used for tapping surface waves from a surface wave delay luie.

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28 SCATTERING OF SH-WAVES

9 9

FiouBE 6. The amplitude A* of the launched surface wave as a function of the single of incidence 0" for

different values of the normalized width 2a/A, of the indentation. The curves I, II, III, IV, V, VI and VH correspond with 2o/Ag = 0.2, 0.4, 0.6, 0.8, 1.0, 1.2 and 1.4, respectively.

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SCATTERING OF SH-WAVES 2 9

FiQUBE 6. The amplitude .d"*^ of the launched surface wave as a function of the angle of incidence 0" for different values of the normalized width 2a/Ag of the discontinuity in mass-loading. The curves I, II,

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In figure 7 the transmission factor T and the reflexion factor R are plotted as a function of the width of the irregularities when a surface wave is incident. From this figure it is observed that \R\ shows a minimum when the width of the triangular indentation is nearly equal to \^ and, in case of a traction-free interruption in the mass-loading, for a width nearly equal to an integral multiple of JAg; no such a minimum occurs for a rigid strip loading. I t is interesting to note that the minima in I iJ I in case of a triangular indentation are located at nearly the same place as the one obtained by Wolf (1970). This author studied the scattered field when a Love wave is incident on a layer having an irregular surface by employing a per-turbation procedure and arrived at minima (of value zero) when the length of the triangular indentation is an integral multiple of the wavelength of the surface wave. The location of the minima in |i?| in case of a discontinuity in mass-loading is in accordance with the predictions based upon a simple transmission line model of the configuration, provided that only surface waves in the configuration are con-sidered. I t is noted that such a simple 'explanation' of the location of the minimum in |.K| in case of the indentation does not apply. For larger values of the normalized width 2a/Ag of the discontinuity in mass-loading, the corresponding values o{ \R\ are in good agreement with those obtained by Quak & Neerhoff (1974), who in-vestigated the scattering of SH-waves by a semi-infinite discontinuity in mass-loading by employing the Wiener-Hopf teclmique. With T/I = 0.2Tt, they obtained

\R\ ~ 0.53 and \R\ ~ 0.15 when the refiexion against a rigid part and a

traction-free part, respectively, were considered. The results presented in figure 7 can, for example, be appUed in micro-electronic circuit design. For example, by choosing the proper surface irregularity, it may be possible to construct phase shifters and filters.

As another appUcation, we mention a study of the scattering properties of a finite diffracting grating delay Une (Tournois & Lagier 1971) by using the scattering properties of a single irregularity, provided that the spacing between the elements of the grating is large enough.

As a (necessary but not sufficient) check on the computations we have used a power relation when a surface wave is incident whereas the reciprocity relation, derived in appendix B, has been used when a bulk wave is incident.

In the computations we have used a combination of single (six significant figures) and double (16 significant figures) precision. A fuUy double-precision program has been used to investigate whether or not round-off errors were significant. I t turned out that the remaining error is determined by the discretization of the Fourier integral representation of the Green function, together with the discretization of the integral equations. An estimate of these errors has been made by increasing the number of intervals into which the contour P+ (cf. (7.2)) is divided and the number of linear equations taken into account to approximate the integral equations.

The computations have been performed on the I.B.M. 360/65 computer of the Computing Centre of the Delft University of Technology. The programs were

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FT

1.0 0.8 0.6 0.4 0.2 0 n 0 —n

"' r^'"

!. ,

f^iwO-'V' ix., 7i=(72=0.2Tt

J-/ "^ / \ - / I \ / \ («) I:6/A5=0.125 II:6/A.=0.25 \ / ' - ' — ~ - S II """••---. 1 1 / — : : - _ - - _ , ^ 0.4 0.8 1.2 1.0 0.8 0.6 0.4 0.2 6 It 0 - T t / 1

\

/

-'7.

"^--41

N 1 1 -- , ^ " - 1 . . 1. 2a „ l;~»iaiii:i;«!;M % Xi o^v^T X, 7,=0.2n (b) I I: rigid (ri^-^oo) Ihtraction-freef^j-O) ' ' l l ' ' ' L 1 r ^ N " " ' s ; — -a4 0.8 1.2 So/A. 2a/Ag

FiouBB 7. The transmission factor T (——) and the reflexion factor S {—) as a function of the normalized width 2a/Ag of the surface irregularities.

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written in the PL/1 language. The average computing time to calculate A+ as a function of ffl together with T and R in one particular configuration is about 9 min for the indentation and about 50 s for a rigid strip loading and a traction-free interruption in the mass-loading together. The resulting accuracy is then better than 2 %. The reason for the discrepancy in computing time between the indenta-tion and the discontinuity in mass-loading follows immediately from the expression (7.2). In case of the indentation, the number of Fourier integrals to be computed is roughly proportional to M^ in view of the factor a;3-l-a;3^ that occurs in (7.2). In case of the discontinuity in mass-loading, however, we have x^+Xs^ = 0 and consequently the number of Fourier integrals then to be computed is proportional to M. {M denotes the number of intervals into which the integration contour L or S is subdivided (cf. §7).)

A P P E N D I X A

In this appendix we determine the Umiting value of a certain type of integral representation for a wave function when the point of observation approaches the relevant path of integration. The Green function occurring in the representation is assumed to satisfy the boundary condition of the third kind on the path of integra-tion which is supposed to be a smooth contour of arbitrary shape. The configuraintegra-tion is shown in figure 8.

FiauEE 8.

In the exterior domain ^ the wave function U' (cf. §2) satisfies the homogeneous Helmholtz equation. With the exception of some part 6^ of the closed contour '8',

U' satisfies the boundary condition of the third kind, i.e. on "ë" —5''we have

(».e„-ffc37?)?7" = 0.

At infinity Ü' satisfies the radiation condition. The Green function G employed in the representation satisfies in S the inhomogeneous Helmholtz equation

(d,d, + ]4)G(x;x^)= -d(x-x^) in S>, (Al)

in which 0 denotes an interior point oi^.Oa^G satisfies the boundary condition

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ia which x ^ x^ and where n^ denotes the unit vector along the normal to 'S

pointing towards ^. Moreover, G satisfies the radiation condition at infinity. Then, appUcation of Green's second identity to the domain Q leads to the representation

U^(Xg') =j^{ü'(s)n,d,G(s;Xg,)-G(8;x^)n,d,U'(8)}d8, (A3)

if ^ is interior to 2. The purpose of this appendix is to determine the Umiting value of (A3) when 0 approaches .5''. As a first step, we write

G(x; Xg,) = ö,(«; x,) + V(x; X,), (A4)

in which G^ denotes the free-space Green function; then, V is regular in ^ as long as 0' is interior to 2. We substitute (A4) in (A3) and subsequently take the Umit of (A3) when 0" approaches some point .2 on .S*" (figure 8). The result is

V'M = WM'r^^{ü'{8)n,b,G^(8;8^)-G^{8;8j)n,d,ü-(s)}d8

+ l i m r {U-(s)n,^,V(8;x^)-V(s;x^)n,^,U-{s)}ds. (A5)

I t is noted that in the derivation of (A 5) we have used the well-known relations Urn f C7«(s)7i,9,É?„(«;«^)d« = i£/»(«^)-f-f £/»(«)».6,öo(s;«j)d« (A6) and lim I G„(«;«^)w^e^t7'(«)d« = i- Go{a;8^)n^d^U'(s)ds. (A7)

a»-».* J y J y

In order to evaluate the remaining limit in (A 5) we first observe that on account of (A4) it follows from (A2) that

(n,S^ + ksV)V(x;x^)= -{n^d^ + ksV)G^(x;x^) on «'. (A8)

By using (A 8) in order to eUminate w„9„F in (A 5) and subsequently re-using (A6) and (A7) we obtain

U'M = - / ^ ^ o ( « ; «j) {n,^. + hv) U'{8)ds

- U m f V(8;Xg,)(n^d, + k3V)U''(8)d8. (A9) In taking the limit Ui (A 9), the arc .^^may be considered as locaUy flat in the neighbourhood of J . Then, with reference to the Green function for a flat boundary (cf. (3.3)) it can be shown that the singularity occurring in F as ^->-.2 is a logarith-mic one, arising from the Hankel function in it. As a consequence, the Umit of the integral occurring in (A 9) leads to a principal value integral in accordance with (A 7). Upon re-using (A 4) we finally obtain from (A 9)

Ü'M = -j-^G(s;8^){n,d, + ksV) U'{s)ds. (AlO)

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A P P E N D I X B

In this appendix we derive a reciprocity relation that exists between the ampli-tude of the launched surface wave when a bulk wave is incident on one hand and the far-field radiation pattern of the scattered bulk wave when a surface wave is incident on the other hand. In elastodynamic theory we have in the case of harmonic time dependence a general reciprocity theorem due to Lord Rayleigh (1873). For our purposes, however, it is more convenient to derive the desired reciprocity relation in a more straightforward manner. This reciprocity relation is obtained from suitable relations that exist between the particle displacements occurring in two different possible situations. We indicate these situations by the subscripts

A and B respectively. Situation A appUes to an incident bulk wave and situation B

appUes to an incident surface wave. First, the desired reciprocity relation is derived for scattering by an indentation (figure 2). As U^ and Ug both satisfy the same boundary condition (2.4) on L, we obtain

I

(U^n, S. Us - Ugn, Ö, U^) ds = 0. (B 1)

L

Next, we apply the reciprocity relation (5.3) with v^= —n^ to the domain D^. In (5.3) we employ the functions W^_B = V\^g. Then, on account of the boundary condition (2.4) on 2 for both t/J^ and (7^, the integral along S vanishes. Hence

ƒ,

(V^ri,d,Uh-U^n^d,U'^)ds = 0. (B 2)

L

Furthermore, we apply (5.3) to the domain D^. Now we use in (5.3) the functions

^A,B — m,fl' i-®- *liö scattered field in the two situations. Then, on the same

grounds as used in §3 (the surface wave contributions to U^B are travelling in the same direction and consequently the integrals along C*ept yield no contribution), it is easily verified that

L

(UJ n, 8, US - Ui n, 9, UJ) ds = 0. (B 3)

f L

Upon substituting U^B = UJ^,B~^A,B in (B 3), using (B 1) and (B 2) and em-plojdng the actual boundary conditions to be satisfied on L for both U^ and Ug, we obtain

f U^i.n, 9, + fcg 7/2) t/A d« = f Ug(n, 9. + fcg rj^) U^ ds, (B 4)

J L J L

for a non-rigid boundary or

f (r>-a 9. U^)U'Bds=\ {n, 9. Ug) U^ ds, (B 5)

J L J L

for a rigid boundary. Next, we substitute in (B 4) and (B 5) the expressions for U^ (cf (2.2), (2.6) and (2.10)) and U^ (cf. (2.2) and (2.8)). Then it is observed from

(5.6) and (5.7) together with (5.1) that the left-hand side in (B 4) and (B 5) is, apart from the factor in front of the integrals in (5.6) and (5.7), nothing but the

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amplitude ^ ^ of the launched surface wave in the case of an incident bulk wave. These amplitudes are of course a function of the angle of incidence Ö', i.e.

A± = A±{d^).

Next, we observe from (6.8) and (6.9) that the right-hand side in (B 4) and (B 5) denotes t h e far-field radiation pattern in the case of an incident surface wave pro-vided that we replace ^ by Ö' in (6.8) and (6.9). Hence, (B 4) and (B 5) lead to

A±(^) = {iksVjkJP^(,f,) - i 7 r < 0 < K (B6)

in which the plus and the minus sign in P± refer to the far-field radiation pattern associated with an incident surface wave travelling in the positive or the negative Xi-direction, respectively. Equation (B 6) constitutes the desired reciprocity relation in the case of an indentation. The reciprocity relation in the case of a discontinuity in mass-loading is derived in a similar way. Replacing L by 2 , w^8„ by dg and using QSUXB = —^aVi^A.Bt (B 4) and (B 5) reduce to

*s 1^ (Vz -Vi)U^ Vh ds = fcg J ^ (V2 - Vi) UB U^ da, (B 7) for a non-rigid discontinuity or

j ^ (^zU^) Uhds = j^(d^Ug) t/i ds, (B 8)

for a rigid strip loading. From this point onward we proceed along exactly the same lines as followed for the indentation. From (5.8) and (5.9), together with (5.1), on one hand and from (6.11) and (6.12) on the other hand, the reciprocity relation (B 6) immediately follows.

The author wishes to thank Professor A. T. de Hoop for his suggestions and remarks concerning the research presented in this paper.

R E E E B E N C E S Abubakar, I . 1963 Proc. Camb. PhU. Soc. 59, 231-248. Aki, K. & L a m e r , K . L . 1970 J. geophya. Res. 75, 933-954.

Alsop, L. E . , G o o d m a n , A. S. & Ash, E . 1971 J. acoust. Soc. Am. 50, 176-180. Gilbert, F . & Knopoff, L. i 9 6 0 J. geophys. Res. 65, 3437-3444.

H a r r i n g t o n , R . F . 1968 Field computation by moment metfwds. N e w Y o r k : T h e MacmiUan C o m p a n y ; L o n d o n : Collier-Macmillan L i m i t e d .

J o n e s , D . S. 1964 The theory 0/ electromagnetism, 449-460. Oxford, L o n d o n , N e w Y o r k , P a r i s : P e r g a m o n Press.

Levy, A. & Deresiewicz, H . 1967 Bull, seismol. Soc. Am. 57, 393—419. Mclvor, I. K . 1969 Bxdl. seismol. Soc. Am. 59, 1349-1364.

Quak, D. & Neerhoff, F . L. 1974 Appl. Sci. Res. 29, 447-460. Rayleigh, L o r d . 1873 London math. Soc. Proc. 4, 357-368. Slavin, L. M . & Wolf, B . 1970 BuU. seismol. Soc. Am. 60, 859-877. Tournois, P . & Lagier, M. 1971 I.E.E.E.Tran^. SU-li, US-lil. Tournois, P . & L a r d a t , C. 1969 I.E.E.E. Trans. SU-16, 107-117. White, R. M. 1970 Proc. I.E.E.E. 58, 1238-1276.

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SCATTERING AND EXCITATION OF

SH-SURFACE WAVES BY A PROTRUSION

AT THE MASS-LOADED BOUNDARY OF AN

ELASTIC HALF-SPACE

FRED L . NEERHOFF Dept. of Electr. Eng., Div. of Electromagnetic Res., Delft Univ. of Technology, Delft, THE NETHERLANDS

Abstract

A rigorous theory of the scattering and excitation of SH-surface waves by a protrusion at the mass-loaded boundary of an elastic half-space is presented. The boundary value problem (which is of the third kind) is solved by employing two suitably chosen Green functions. One of them is represented as a Fourier type of integral, the other is taken to be the Bessel function of the second kind and order zero. The procedure leads to a system of three, coupled, integral equations. This system is solved numerically. In case of an incident bulk wave, the amplitude of the launched surface wave is computed; in case of an incident surface wave, its transmission and reflection factor are computed. For both cases, an expression for the far-field radiation pattern of the scattered bulk wave is derived. A reciprocity relation is shown to exist between the amplitude of the launched surface wave and the far-field bulk wave radiation pattern. Numerical results are presented for a triangularly-prismatic protrusion; they are compared with the results pertaining to a corresponding indentation in the mass-loaded boundary, that have been obtained in a previous paper.

§ 1. Introduction

The subject of investigation of the present paper is the scattering and excitation of SH-surface waves in a mass-loaded elastic half-space on which a protrusion is present. The problem is solved with the aid of an extension of the method employed in a previous paper [1], where the scattering by an indentation and a plane, finite discontinuity in mass-loading are considered. The scattering by a semi-infinite discontinuity in mass-loading has been treated in [2].

It has been noted (see, for example [3]) that the mass-loaded boundary is a good low-frequency approximation for describing the Love wave

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SCATTERING AND EXCITATION OF SH-SURFACE WAVES

configuration. Consequently, our results are of importance in connection

with the scattering and excitation of Love waves, especially where this

type of surface wave finds application in micro-electronics (see also [4]).

The resulting boundary-value problem (which is of the third kind) for

the configuration under investigation is formulated in section 2. It is

solved by employing two suitably chosen Green functions: one of them

is taken to be the half-space Green function and is represented as a

Fourier type of integral, the other is taken to be the Bessel function of

the second kind and order zero; its domain of definition being the (finite)

domain, occupied by the protrusion (section 3). In section 4 we derive a

system of three, coupled, integral equations in which the relevant field

distributions on the boundaries of the protrusion occur as unknown

quantities. In section 5, the launched surface wave amplitude in the case of

an incident bulk wave and the transmission and reflection factor in the case

of an incident surface wave are expressed in terms of these distributions.

Furthermore, for both cases, an expression is derived for the far-field

radiation pattern of the scattered bulk wave. Once the method of solution

of the resulting system of integral equations has been discussed (section 6),

numerical results for a triangularly-prismatic protrusion are presented

(section 7). These results are compared with those pertaining to a

corre-sponding indentation in the mass-loaded boundary, which have been

obtained in [1].

In the Appendix, we derive a reciprocity relation that exists between

the amplitude of the launched surface wave in the case of an incident

bulk wave, and the far-field radiation pattern of the scattered bulk wave

in the case of an incident surface wave.

§ 2. Formulation of the problem

The position in space is specified by the right-handed Cartesian

coordi-nates Xi, JC2 ^nd Xj. We consider a mass-loaded, elastic medium occupying

the half-space — oo < Xi < oo, — oo < jcj < oo, 0 < Xj < oo. The

surface mass density of the applied loading is constant and is denoted

by (Tj. Along the surface, a protrusion is present; it is assumed to be

cylindrical in the X2-direction and of bounded extent in the Xj-direction.

The surface mass density 02 of the loading applied on it may differ from

ffj and vary with position. The bulk medium is assumed to be linear,

isotropic, homogeneous and perfectly elastic. Its mechanical properties

are characterized by the Lamé coefficients A and /x and the volume

mass-density p (Fig. 1).

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SCATTERING AND EXCITATION OF SH-SURFACE WAVES 39

Fig. 1. The protrusion and incident waves.

The incident SH-wave is taken to be time harmonic and propagating

in the x^, jC3-plane. The complex time factor exp( —icot), where co denotes

the circular frequency and t the time, is omitted throughout.

Further-more, the summation convention is used for the repeated subscript a;

it is to be assigned the values 1 and 3. The A:2-component of the particle

displacement (which is the only non-vanishing component) is denoted by

[/^'^ in the half-space and by t/'^^ in the domain occupied by the

pro-trusion. We write

l/<'>(jc) •= U''(x) + U'(x), (2.1)

in which x = (x^, x^) and where l/' is defined by

f t^Bw + t^Bw when a bulk wave is incident,

j ^ , _ 1 BW BW ^2 2)

[U's^ when a surface wave is incident.

In (2.2),

Ï7BW

denotes the incident bulk wave, t/ew is the bulk wave

reflected against the plane boundary of the half-space with uniform

mass-loading and U'sw denotes the incident surface wave, travelling in the

positive or the negative Xi-direction, respectively (Fig. 1). The wave

function U' introduced through (2.1) is denoted as the scattered field.

In its domain of definition, each of the wave functions satisfies the

source-free elastodynamic wave equation for SH-waves, viz.

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40 SCATTERING AND EXCITATION OF SH-SURFACE WAVES

where d^ denotes the partial derivative with respect to x^, and ^s 's the wave number of shear waves {k^ — w(p/n)^).

At the mass-loaded boundary, the following boundary condition has to be satisfied [1]

(«,5j + k^t])U = 0 at a mass-loaded boundary, (2.4) in which «^ denotes the unit vector in the direction of the normal to the boundary as shown in Fig. I, while r] = k^a/p denotes the reduced specific acoustic impedance of the massloading (rj = /;, when CT = CT, and rj = ri2 when a = o-j). Furthermore, the particle displacement and the normal component of the stress tensor are continuous across the interface I (Fig. 1). This yields

l/d) = 1/(2) and M^jt/'" = lidjU^^^ at I . (2.5) The incident waves [/éw and L/j^ are taken to be plane waves of unit amplitude and zero phase at the origin. Hence

U B W W = exp(iasJri - iysXj), (2.6)

where a^ = kss'm(9') and y'^ = ksC0s{9'), in which 0', with —-^n <

< 0' < ^n, denotes the angle of incidence as shown in Fig. 1. The

incident surface wave is given by

Us^x) = e\p{±\k„Xi - Ml.^3). (2.7)

in which k^ = k^(\ + r}^)^ > 0 denotes the wave number of the surface wave propagating in the A:i-direction.

Finally, the scattered field U^ should satisfy the radiation condition at infinity. This implies that at large distances from the disturbance, (7* consists of cylindrical bulk waves and plane surface waves, both travelling away from the protrusion.

§ 3. Integral representations

In this section we derive integral representations for the scattered field

U' and the total field L/*^'. To that aim, we employ two suitably chosen

Green functions, to be denoted by G*'' and C*^'. The undisturbed half-space is taken as the domain of definition of C ' " , whereas the (finite) domain occupied by the protrusion, is taken as the domain of definition of G(^>.