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Ultrasonics 46 (2007) 235–250

DPSM technique for ultrasonic field modelling near fluid–solid interface

Sourav Banerjee a, Tribikram Kundu a,*, Nasser A. Alnuaimi b

a Department of Civil Engineering and Engineering Mechanics, University of Arizona, Tucson, AZ 85721, USAb College of Engineering, Civil Engineering Department, University of Qatar, Doha, Qatar

Received 15 February 2007; accepted 17 February 2007Available online 24 February 2007

Abstract

Distributed point source method (DPSM) is gradually gaining popularity in the field of non-destructive evaluation (NDE). DPSM is asemi-analytical technique that can be used to calculate the ultrasonic fields produced by transducers of finite dimension placed in homo-geneous or non-homogeneous media. This technique has been already used to model ultrasonic fields in homogeneous and multi-layeredfluid structures. In this paper the method is extended to model the ultrasonic fields generated in both fluid and solid media near a fluid–solid interface when the transducer is placed in the fluid half-space near the interface. Most results in this paper are generated by thenewly developed DPSM technique that requires matrix inversion. This technique is identified as the matrix inversion based DPSM tech-nique. Some of these results are compared with the results produced by the Rayleigh–Sommerfield integral based DPSM technique. The-ory behind both matrix inversion based and Rayleigh–Sommerfield integral based DPSM techniques is presented in this paper. Thematrix inversion based DPSM technique is found to be very efficient for computing the ultrasonic field in non-homogeneous materials.One objective of this study is to model ultrasonic fields in both solids and fluids generated by the leaky Rayleigh wave when finite sizetransducers are inclined at Rayleigh critical angles. This phenomenon has been correctly modelled by the technique. It should be men-tioned here that techniques based on paraxial assumptions fail to model the critical reflection phenomenon. Other advantages of theDPSM technique compared to the currently available techniques for transducer radiation modelling are discussed in the paper underIntroduction.� 2007 Elsevier B.V. All rights reserved.

Keywords: DPSM technique; Fluid–solid interface modelling; Critical angle; Leaky wave; Rayleigh-wave

1. Introduction

Modeling of ultrasonic and sonic fields generated byplanar transducers of finite dimension is one of the basicproblems in textbooks in this area [1–4]. A good reviewof the earlier developments of the ultrasonic field modelingin front of a planar transducer can be found in Ref. [5]. Alist of more recent developments in this field of research hasbeen given by Sha et al. [6]. The pressure field in front of aplanar transducer in homogeneous isotropic materials hasbeen computed both in the time domain [5,7,8] and in thefrequency domain [9–15]. In addition to the ultrasonic field

0041-624X/$ - see front matter � 2007 Elsevier B.V. All rights reserved.

doi:10.1016/j.ultras.2007.02.003

* Corresponding author. Tel.: +1 520 621 6573; fax: +1 520 621 2550.E-mail addresses: sourav@email.arizona.edu (S. Banerjee), tkundu@

email.arizona.edu (T. Kundu), anasser@qu.edu.qa (N.A. Alnuaimi).

modeling in isotropic materials progress has been made inthe modeling of the ultrasonic radiation field in trans-versely isotropic and orthotropic media also [16,17]. Mostof the above mentioned investigations are based on Huy-gen’s principle where the total field is obtained from the lin-ear sum of point sources distributed over the transducer.The integral representation of this field is known as Ray-leigh–Sommerfield integral. Since 2-D numerical integra-tion on the transducer aperture is a time consumingoperation Lerch et al. [14] proposed the edge element tech-nique that requires 1-D line integrals only. Wen and Bre-azeale [18] proposed an alternative approach. Theycomputed the total field by superimposing a number ofGaussian beam solutions. They have shown that by super-imposing only 10 Gaussian solutions the field radiated by acircular piston transducer can be modeled. Schmerr [19]

236 S. Banerjee et al. / Ultrasonics 46 (2007) 235–250

followed this approach to compute the ultrasonic field neara curved fluid–solid interface. Later Spies [20] and Schmerret al. [21] extended this technique to a homogeneous aniso-tropic solid and water immersed anisotropic solid, respec-tively. Another technique based on Gauss–Hermite beammodel for ultrasonic field modeling in anisotropic materialswas proposed by Newberry and Thompson [22].

Although a significant progress has been made in theultrasonic field modeling in a homogeneous medium, theeffect of an interface near an ultrasonic transducer of finitedimension has not been studied extensively yet. Lerch et al.[15] applied the edge element technique to model the ultra-sonic fields near a fluid–solid interface. Schmerr [19] andSchmerr et al. [21] studied the ultrasonic field near afluid–solid curved interface using multi-Gaussian beammodeling technique. Spies [23] also used this technique tostudy the ultrasonic wave propagation in orthotropic lay-ered composites with far-field approximation where nofluid–solid interface effects were addressed. Although themulti-Gaussian beam modeling technique has some com-putational advantage it also has a number of limitationssimilar to those of other paraxial models. For example, itcannot correctly model the critical reflection phenomenon;it cannot model a transmitted beam at an interface neargrazing incidence. This technique also fails if the interfacehas different curvatures, or when the radius of curvatureof the transducer is small, as observed in acoustic micros-copy experiments with its tightly focused lens. A detaildescription of the limitations of the multi-Gaussian parax-ial models can be found in Schmerr et al. [21]. A newlydeveloped technique called DPSM (distributed pointsource method), proposed by Placko and Kundu [24,25]avoids the above-mentioned limitations and not requireany far field approximation. In this technique one layerof point sources are distributed near the transducer faceand two layers are placed near the interface. Point sourcesare not placed on the transducer aperture or interface sur-faces to avoid the singular terms. This concept of placingpoint sources near the boundary is similar to the indirectboundary element method that uses auxiliary boundary.The advantage of the matrix inversion based DPSM tech-nique is that it not only avoids the paraxial approximationit also does not require any ray tracing or transmission andreflection coefficient computation at the interface. Allmethods developed so far for ultrasonic field radiationmodeling near a fluid–solid interface require computationof reflection and transmission coefficients at the interface[15,19,21,26]. Ray tracing and reflection/transmission coef-ficient computation become cumbersome in presence ofmultiple interfaces while such geometries can be relativelyeasily modeled by the matrix inversion based DPSM tech-nique [27].

DPSM technique for ultrasonic field modeling was firstdeveloped by Placko and Kundu [24]. They successfullyused this technique to model ultrasonic fields in a homoge-neous fluid, and in a non-homogeneous fluid with oneinterface [28] and multiple interfaces [27]. The interaction

between two transducers, for different transducer arrange-ments and source strengths, placed in a homogeneous fluidhas been studied by Ahmad et al. [29]. The scattered ultra-sonic field generated by a solid scatterer of finite dimensionplaced in a homogeneous fluid has also been modeled bythe DPSM technique [30]. Recently the method has beenextended to model the phased array transducers [31]. Allthese works modeled the ultrasonic field in a fluid medium.In this paper, a fluid–solid half-space structure with aninterface is modeled for critical and non-critical angles ofincidence. The theory behind two different techniques ofDPSM is presented here. The first one is based on the con-ventional Rayleigh–Sommerfield integral technique and thesecond one is newly developed Matrix inversion basedDPSM technique. Traditional surface integral techniqueor Rayleigh–Sommerfield integral technique has beenadopted here for comparison purposes only to validatethe matrix inversion based DPSM technique. It should benoted here that the traditional Rayleigh–Sommerfield inte-gral based DPSM technique is to some extent similar toother available techniques based on the point source super-position [23,26] that require a number of point sources dis-tributed near or on the transducer face and thentransmission and reflection coefficients at the interface needto be computed. Many modeling techniques such as multi-Gaussian beam models and other paraxial models availabletoday find it difficult to model leaky guided waves. How-ever, leaky Rayleigh waves along the fluid–solid interfacecan be easily modeled by the matrix inversion based DPSMtechnique presented here.

2. Theory

DPSM is briefly described in Section 2.1 then detailmathematical derivations are presented in the followingsections.

2.1. Distributed point source method

If the front face of a transducer is considered as the mainsource of an ultrasonic field, then the ultrasonic field gen-erated by that source can be assumed to be the summationof the ultrasonic fields generated by a number of pointsources distributed near that finite source. Any interfaceis responsible for generating reflected and transmittedultrasonic fields. Therefore, the interface can be replacedby two layers of sources – one layer generating the reflectedfield and the second layer generating the transmitted field.Two layers of interface sources are distributed on two sidesof the interface. Strengths of the point sources distributednear the transducer face and the interface are obtainedby satisfying the boundary conditions and interface conti-nuity conditions. For solving this problem we need pointsource solutions for both solid and fluid media. Therefore,the first step is to calculate the stress and displacementGreen’s functions in the solid, and pressure and displace-ment Green’s functions in the fluid.

S. Banerjee et al. / Ultrasonics 46 (2007) 235–250 237

2.2. Point source excitation in a solid

Our first objective is to obtain the displacements andstresses in a solid due to a point source excitation, or inother words, the Green’s functions for the solid. When apoint source is acting in a solid, the body force will changeto a concentrated time dependent impulsive force. Thisforce can be represented by Dirac delta function in space.So, decoupling the independent parameters, the body forcecan be written as

Fðx; tÞ ¼ P � f ðtÞdðxÞ or F i ¼ P if ðtÞdðxjÞ ð1Þwhere P is the force vector without the time and spacedependence.

2.3. Displacement Green’s function

It can be shown that for the body force given in Eq. (1),when the point source acts at y then the displacement fieldat location x is given by [32,33],

ui ¼ U ie�ixt ¼ Gijðx; yÞP je

�ixt ð2ÞHere Gij(x;y) is called the space dependent Green’s func-tion of displacement for the isotropic homogeneous solid.Substituting r = jx � yj, the displacement Green’s functioncan be written as

Gijðx; yÞ ¼ 1

4pqx2

eikpr

rk2

pRiRj þ ð3RiRj � dijÞikp

r� 1

r2

� �� ��þ eiksr

rk2

s ðdij � RiRjÞ � 3RiRj � dij

� � iks

r� 1

r

� �� ��;

where Ri ¼xi � yi

rð3Þ

In the matrix form

Gðx;yÞ¼ G1ðx;yÞ G2ðx;yÞ G3ðx;yÞ� T

and u¼Gðx;yÞPð4Þ

If the unit excitation force at y acts in the jth direction, thenthe displacement at x in the ith direction is given byGij(x;y).

2.4. Calculation of stress Green’s function

For isotropic homogeneous solids the expression forstresses can be written as

rij ¼ 2leij þ kdijekk ð5Þwhere k, l are the two Lame constants and dij is the Kro-necker delta. We know that strain can be expressed as afunction of displacement, eij ¼ 1

2ðui;j þ uj;iÞ. Substituting

the expression for displacement (Eq. (2)) in the expressionfor strains

eij ¼1

2ðGik;j þ Gjk;iÞP k ð6Þ

In the following the harmonic time dependence is impliedand is not explicitly written for convenience. Substituting

the expression for strains in Eq. (5), the stress Green’s func-tion at x due to a concentrated time harmonic force at y

can be obtained. For isotropic homogeneous linearly elas-tic material, we can write the expression for the stressGreen’s function at x due to a concentrated harmonic forceat y as

Sijðx; yÞ ¼ lðGik;j þ Gjk;iÞP k þ kdijGkq;kP q ð7Þor Sijðx; yÞ ¼ ðlðGik;j þ Gjk;iÞdkq þ kdijGkq;kÞP q ð8Þ

By rigorous differentiation, expressions for all stresses fromEq. (7) have been found and presented in Appendix 1.

2.5. Computation of displacements and stresses in the solid

due to a group of point sources

When a group of point sources distributed over a finitedimension in a solid are excited, the response at any pointin the solid can be computed by superimposing the contri-bution of each point source. Let there be a solid half spaceand M number of point sources at the free boundary,above and below the interface, as shown in Fig. 1. Ourobjective is to compute the response at point C due to M

number of point sources at the interface I. Response at C

will be the superposition of the contribution of each pointsource. So, the total displacement at C can be written as

ui ¼XM

m¼1

ðGmi1P m

1 þ Gmi2P m

2 þ Gmi3P m

3 Þ ¼XM

m¼1

GimPm ð9Þ

Similarly the stresses at C may be written as

Sij ¼XM

m¼1

½ðr1ijÞ

mP m1 þ ðr2

ijÞmP m

2 þ ðr3ijÞ

mP m3 � ¼

XM

m¼1

sijm P

4p

� �m

ð10Þ

Equations are written in conventional index notationwhere i, j takes values 1, 2 and 3. rp

ij has been defined inAppendix 1 in Eqs. (A.35)–(A.38)).

2.6. Matrix representation

In our earlier discussion, the steps to compute the dis-placements and stresses at a point have been presented.The equations can be written in the matrix form. Let usnow compute the displacement and stress fields at a groupof observation points (we will call them target points). Weassume that there are N number of target points and thateach displacement and stress expression at N target pointsdue to M source points are to be computed. N target pointsand M source points are shown in Fig. 1.

One can write,

uiT ¼ DSiTS � AS ð11ÞsijT ¼ SijTS � AS ð12Þ

Fig. 1. A schematic diagram of M source points and N target points in solid and fluid of a fluid–solid structure.

238 S. Banerjee et al. / Ultrasonics 46 (2007) 235–250

where

fASgT¼ 1

4pP1 P2 P3 P4 P5 . . . . . . PM�2 PM�1 PM� T

ð3M�1Þ

ð13Þ

and i, j take values 1, 2 and 3. Subscripts T and S representtarget and source points, respectively.

Pi is a (1 · 3) vector, defined in Appendix 1 (Eq. (A.40)).Stress and displacement matrices of Eqs. (11) and (12) havethe following form. Only S33TS and DS3TS matrices aregiven below

S33TS ¼

s1133 s21

33 s3133 s41

33 s5133 . . . . . . sM�21

33 sM�1133 sM1

33

s1233 s22

33 s3233 s42

33 s5233 . . . . . . sM�22

33 sM�1233 sM2

33

s1333 s23

33 s3333 s43

33 s5333 . . . . . . sM�23

33 sM�1333 sM3

33

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

s1N�233 s2N�2

33 s3N�233 s4N�2

33 s5N�233 . . . . . . sM�2N�2

33 sM�1N�233 sMN�2

33

s1N�133 s2N�1

33 s3N�133 s4N�1

33 s5N�133 . . . . . . sM�2N�1

33 sM�1N�133 sMN�1

33

s1N33 s2N

33 s3N33 s4N

33 s5N33 . . . . . . sM�2N

33 sM�1N33 sMN

33

2666666666666664

3777777777777775ðN�3MÞ

ð14Þ

and

DS3TS¼

G113 G21

3 G313 :: GM�11

3 GM13

G123 G22

3 G323 . . . GM�12

3 GM23

G133 G23

3 G333 . . . GM�13

3 GM33

. . . . . . :: . . . :: . . .

G1N�13 G2N�1

3 G3N�13 . . . GM�1N�1

3 GMN�13

G1N3 G2N

3 G3N3 . . . GM�1N

3 GMN3

26666666664

37777777775ðN�3MÞ

ð15Þ

smn33 and Gmn

3 at the nth target point due to mth point sourcehave the following form:

smn33 ¼ ðr1

33Þm ðr2

33Þm ðr3

33Þm�

nand Gmn

3 ¼ Gm31 Gm

32 Gm33½ �nð16Þ

Similarly, other stress and displacement matrices can alsobe written in this form.

2.7. Pressure and displacement Green’s functions in the fluid

A perfect fluid is defined as a homogeneous isotropicmedium that has no shear strength and the pressure at a

point in all directions in the fluid must be the same. Thepressure Green’s function (Gf) in fluid at x due to a pointsource acting at y [3,4] can be written as

Gfðr;xÞ ¼eikf r

4prð17Þ

where

kf ¼xcf

S. Banerjee et al. / Ultrasonics 46 (2007) 235–250 239

and r ¼ jx� yj ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðx1 � y1Þ

2 þ ðx2 � y2Þ2 þ ðx3 � y3Þ

2q

.The pressure in the fluid can also be represented by a po-tential function [32]. The pressure-potential and the dis-placement-potential relations can be written as

p ¼ �qx2/f ð18Þ

ui ¼o/f

oxið19Þ

where /f is the scalar potential. Therefore, for the pressureGreen’s function, the potential function can be expressedas

/fðr;xÞ ¼ �eikf r

4pqx2rð20Þ

Taking derivatives of /f with respect to xi, the displacementcomponents in the three directions can be obtained

ui ¼1

4pqx2

1

rikf Rie

ikf r � eikf r

r2Ri

� �ð21Þ

where

Ri ¼xi � yi

rand i takes value 1; 2 and 3:

2.8. Computation of displacement and pressure in a fluid

generated by a group of point sources

Combined effect of a large number of point sources dis-tributed over a plane surface such as the transducer face isthe vibration of particles in a direction normal to the planesurface. The total pressure field can be written in the sum-mation form [4]

pðxÞ ¼XN

m¼1

B4p

DSm

� �expðikfrmÞ

rm¼XN

m¼1

AmexpðikfrmÞ

rmð22Þ

In the DPSM technique it has been assumed that eachpoint source distributed over the surface has a differentsource strength as specified by Am, where m designatesthe mth point source and rm is the distance of the targetpoint x from the mth point source. Hence, pressure atany point at a distance rm from the mth point source withsource strength Am can be written as

pmðrÞ ¼ AmexpðikfrmÞ

rmð23Þ

and due to N number of point sources distributed on a sur-face, the pressure at the target point is

pðxÞ ¼XN

m¼1

pmðrmÞ ¼XN

m¼1

Amexpðikf rmÞ

rmð24Þ

From the pressure velocity relation, it is also possible toobtain the velocity in all three directions at x due to N

number of point sources placed at y.

� opon¼ q

ovn

ot¼ �ixqvn ð25Þ

where opon is the derivative of pressure along the direction n.

The velocity can be written as

vn ¼1

ixqopon

ð26Þ

Therefore, the velocity in the radial direction, at a distancer from the mth point source, is given by

vmðrÞ ¼Am

ixqo

orexpðikfrÞ

r

� �¼ Am

ixqikf expðikf rÞ

r� expðikfrÞ

r2

� �¼ Am

ixqexpðikfrÞ

rikf �

1

r

� �ð27Þ

and the three components of velocity are

vjmðrÞ ¼Am

ixqo

oxj

expðikf rÞr

� �¼ Am

ixqxj expðikf rÞ

r2ikf �

1

r

� �ð28Þ

where j takes values 1, 2 and 3. When the contributions ofall N sources are added, the total velocity in x1, x2 and x3

directions at point x can be written

vjðxÞ ¼XN

m¼1

vjmðrmÞ ¼XN

m¼1

Am

ixqxjm expðikfrmÞ

r2m

ikf �1

rm

� �ð29Þ

where xjm is the shortest distance along xj direction be-tween the mth point source and the target point, as shownin Fig. 2a. If the transducer surface is parallel to the x1x2-plane and its velocity in the x3 direction is given by v0 thenfor all x values on the transducer surface the velocityshould be equal to v0. Therefore,

v3ðxÞ ¼XN

m¼1

Am

ixqx3m expðikf rmÞ

r2m

ikf �1

rm

� �¼ v0 ð30aÞ

and if the transducer face is inclined at an angle of h, whenrotated about the x2 axis (Fig. 1), the velocity of the trans-ducer face can be expressed as

v1ðxÞSinhþ v3ðxÞCosh

¼XN

m¼1

Am

ixqikf �

1

rm

� �x1m expðikf rmÞ

r2m

Sinhþx3m expðikf rmÞr2

m

Cosh

� �¼ v0 ð30bÞ

2.9. Matrix representation

Velocity of the M target points placed on the transducerface due to point sources distributed just below the trans-ducer surface at a distance rS [4], can be written in matrixform as

Fig. 2. (a) Schematic diagram of an observation point and mth point source on the transducer face. (b) Side view of a transducer and location of pointsources on its face.

240 S. Banerjee et al. / Ultrasonics 46 (2007) 235–250

VS ¼MSSAS ð31Þwhere VS is the (M · 1) vector of the velocity components,perpendicular to the transducer surface. If the velocityof the transducer face is given by v0, then VS can bewritten as

fVSgT¼ v10 v2

0 v30 . . . . . . . . . . . . . . . vM�1

0 vM0

� T ð32Þ

where vn0 is the velocity of the nth target point. If AS is the

(N · 1) vector of the source strengths, then

fASgT¼ A1 A2 A3 A4 A5 A6 . . . . . . AN�2 AN�1 AN½ �T

ð33ÞFrom the earlier discussion, we know that each pointsource is placed inside a sphere and hence, the numberof apex points of the spheres touching the transducersurface will be the same as the number of point sources.The optimum distance (rS) of the point sources from thetransducer face and the interface has been given byPlacko and Kundu [33]. When the target points areplaced at the apex of the spheres of the point sources,then M is equal to N. Therefore, for the target pointsat the apex of the spheres of the point sources, the squarematrix MSS can be written as

MSS ¼

f ðx1t1; r

11Þ f ðx2

t1; r21Þ f ðx3

t1; r32Þ f ðx4

t1; r41Þ . . . . .

f ðx1t2; r

12Þ f ðx2

t2; r22Þ f ðx3

t2; r32Þ f ðx4

t2; r42Þ . . . . .

f ðx1t3; r

13Þ f ðx2

t3; r23Þ f ðx3

t3; r33Þ f ðx4

t3; r43Þ . . . . .

f ðx1t4; r

14Þ f ðx2

t4; r24Þ f ðx3

t4; r34Þ f ðx4

t4; r44Þ . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

f ðx1tN ; r

1N Þ f ðx2

tN ; r2N Þ f ðx3

tN ; r3N Þ f ðx4

tN ; r4N Þ . . . . .

266666666666664

where

f ðxmtn; r

mn Þ ¼

xmtn expðikfrm

n Þixqðrm

n Þ2

ikf �1

rmn

� �¼ expðikfrm

n Þixqðrm

n Þ2

� ikf �1

rmn

� �ðxm

3n Coshþ xm1n SinhÞ ð35Þ

and rmn is the distance between the mth point source and the

nth target point.For a general set of target points located on any sur-

face, the velocity due to the transducer sources can bewritten as

VT ¼MTSAS ð36Þ

where VT, the velocity vector (M · 1) contains the normalvelocity components of the target points distributed onthe surface. The matrix MTS has elements that are similarto those of MSS, with different xm

tn values and the size ofthe matrix is (M · N), where M is the number of targetpoints and N is the number of source points. Followingthe same concept, the pressure at any M number of targetpoints due to N number of source points can be written as

. f ðxN�1t1 ; rN�1

1 Þ f ðxNt1; r

N1 Þ

. f ðxN�1t2 ; rN�1

2 Þ f ðxNt2; r

N2 Þ

. f ðxN�1t3 ; rN�1

3 Þ f ðxNt3; r

N3 Þ

. f ðxN�1t4 ; rN�1

4 Þ f ðxNt4; r

N4 Þ

. . . . . . .

. . . . . . .

. . . . . . .

. f ðxN�1tN ; rN�1

N Þ f ðxNtN ; r

NN Þ

377777777777775N�N

ð34Þ

S. Banerjee et al. / Ultrasonics 46 (2007) 235–250 241

PRT ¼ QTSAS ð37Þ

where PRT is the (N · 1) vector of pressure values at N tar-get points, and QTS is a (N · M) matrix given below

QTS ¼

expðikf r11Þ

r11

expðikf r21Þ

r21

expðikf r31Þ

r31

. . . . . .expðikf rM

1Þ

rM1

expðikf r12Þ

r12

expðikf r22Þ

r22

expðikf r32Þ

r32

. . . . . .expðikf rM

2Þ

rM2

expðikf r13Þ

r13

expðikf r23Þ

r23

expðikf r33Þ

r33

. . . . . .expðikf rM

3Þ

rM3

. . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .expðikf r1

N Þr1

N

expðikf r2N Þ

r2N

expðikf r3N Þ

r3N

. . . . . .expðikf rM

N ÞrM

N

26666666666664

37777777777775N�M

ð38Þ

When the target points are located at the apex of thespheres of the point sources, Eq. (37) takes the form

PRS ¼ QSSAS ð39Þwhere QSS is a (N · N) matrix.

The definition of rmn is identical to that given in Eq. (35).

It is the distance between the mth point source and the nthtarget point.

In the same manner, The matrix expression for displace-ments at general target points in the fluid can be written as

UiT ¼ DFiTSAS ð40Þwhere

DFiTS ¼

gðR1i1; r1

1Þ gðR2i1; r2

1Þ gðR3i1; r3

1Þ . . . gðRM�1i1

; rM�11 Þ gðRM

i1; rM

1 ÞgðR1

i2; r1

2Þ gðR2i2; r2

2Þ gðR3i2; r3

2Þ . . . gðRM�1i2

; rM�12 Þ gðRM

i2; rM

2 ÞgðR1

i3; r1

3Þ gðR2i3; r2

3Þ gðR3i3; r3

3Þ . . . gðRM�1i3

; rM�13 Þ gðRM

i3; rM

3 ÞgðR1

i4; r1

4Þ gðR2i4; r2

4Þ gðR3i4; r3

4Þ . . . gðRM�1i4

; rM�14 Þ gðRM

i4; rM

4 Þ. . . . . . . . . . . . . . . . . .

gðR1iN; r1

N Þ gðR2iN; r2

N Þ gðR3iN; r3

N Þ . . . gðRM�1iN

; rM�1N Þ gðRM

iN; rM

N Þ

26666666664

37777777775ðN�MÞ

ð41Þ

where gðRmin; rm

n Þ ¼1

qx2

1

rmn

ikf Rmin

eikf rmn � eikf rm

n

ðrmn Þ

2Rm

in

" #ð42Þ

Rmin¼

xmin� ym

in

rmn

and i ¼ 1; 2; 3

2.10. Ultrasonic field modeling near a fluid–solid interface byDPSM matrix inversion technique

Let us consider a plane interface between a solid half-space and a fluid half-space analogous to a river bed withwater. A schematic diagram of the system considered forour analysis is shown in Fig. 1, where fluid half-space isshown below the solid half-space. Only a few point sourcesare shown along the interface in the diagram to keep it sim-

ple; however in the actual model the point sources are dis-tributed over the entire interface.

There is a circular transducer immersed in the fluid. Anumber of point sources are distributed below the trans-ducer face and on both sides of the interface. These sourceswhen superimposed, should produce the total ultrasonicfield in fluid and solid media. AI is the source strength ofthe point sources that are placed above the solid–fluidinterface and generate the reflected ultrasonic field in thefluid. Similarly, A�I is the source strength of the sources thatare distributed below the solid–fluid interface and modelthe transmitted field in the solid. The point sources thathave been distributed below the transducer face have thesource strength vector AS. We intend to compute the ultra-sonic field in both solid and fluid media. In Fig. 1, twopoints (C and D) have been considered for the illustrationpurpose. The ultrasonic field at point C is the summationof the contributions of all point sources distributed belowthe interface. Similarly, the Ultrasonic field at point D willbe the summation of contributions of the point sources dis-tributed above the solid-fluid interface and below the trans-ducer face.

2.11. Matrix formulation to calculate source strengths

The velocity and pressure fields in the fluids can beexpressed in the matrix form when we take a mesh-gridof target points in the solid and fluid media as shown inFig. 1. Referring to Eq. (31) one can write

VS ¼MSSAS ð43Þwhere VS is the (N · 1) vector of the velocity components atN number of source points distributed below the trans-ducer face and AS is the (N · 1) vector containing thestrength of the transducer sources. The elements of MSS

has been presented in Eqs. (34) and (35); this matrix is de-fined when all target points are distributed on the surfaceof the transducer. For a more general case when targetpoints are not necessarily on the transducer surface, thevelocity due to transducer sources can be written as

VT ¼MTSAS ð44Þ

and due to interface sources as

VT ¼MTIAI ð45Þ

242 S. Banerjee et al. / Ultrasonics 46 (2007) 235–250

when the interface has M source points distributed on eachside of the interface. Therefore, A1 has (M · 1) elements.Similarly the pressure at any set of target points in the fluiddue to transducer sources can be written as

PRsT ¼ QTSAS ð46Þ

and due to interface sources, the pressure at the same set oftarget points in the fluid can be written as

PR1T ¼ QTIAI ð47Þ

Therefore, at those target points the total pressure field dueto both sets of sources will be

PRT ¼ PRsT þ PR1

T ¼ QTSAS þQTIAI ð48Þ

Similarly, at any set of target points, the displacementalong the x3 direction in the fluid can be written as

U3T ¼ DF3TSAS þDF3TIAI ð49Þ

Every point source that has been considered to calculatethe transmitted field in solid, has three different pointforces in three different directions.

The stress along the x3 direction at the interface can bewritten from Eq. (12) as

s33II� ¼ S33II�A�II ð50Þ

Similarly, for shear stresses, from Eq. (12), we can write

s31II� ¼ S31II�A�II ð51Þ

s32II� ¼ S32II�A�II ð52Þ

where A�I is the source strength vector of the sources dis-tributed above the fluid–solid interface and it has 3Melements.

Fig. 3. Ultrasonic transducer in front of a fluid–solid interface –transmitted P and S waves in the solid and reflected P-wave in the fluidare shown.

2.12. Boundary conditions

Across the fluid–solid interface the displacement normalto the interface should be continuous. Also, at the inter-face, the compressive normal stress (-s33) in the solid andthe pressure in the fluid should be continuous. The shearstress in the solid at the interface should vanish. If thevelocity of the transducer face is assumed to be VS0 wecan write the boundary conditions as On the transducersurface

MSSAS þMSIAI ¼ VS0 ð53ÞOn the interface, from the continuity of the normal stress,

QISAS þQIIAI ¼ �S33II�A�I ð54Þ

Continuity of the normal displacement gives

DF3ISAS þDF3IIAI ¼ DS3II�A�I ð55Þ

and from the vanishing shear stress condition at the fluid–solid interface

S31II�A�I ¼ 0 ð56Þ

S32II�A�I ¼ 0 ð57Þ

Eqs. (53) to (57) can be written in matrix form

MSS MSI 0

QIS QII S33II�

DF3IS DF3II �DS3II�

0 0 S31II�

0 0 S32II�

26666664

37777775ðNþ4MÞ�ðNþ4MÞ

AS

AII

A�II

8><>:9>=>;ðNþ4MÞ

¼VS0

0

0

8><>:9>=>;

Nþ4M

ð58Þor

½MAT�fKg ¼ fVg ð59ÞThe vector of source strengths of the total system can becalculated from the above equation

fKg ¼ ½MAT��1fVg ð60ÞAfter calculating the source strengths, the pressure, veloc-ity, stress and displacement values at any point can beobtained.

2.13. Rayleigh–Sommerfield integral technique or surface

integral technique

Pressure field in a fluid for a planar piston transducer offinite diameter can be obtained by superimposing the basicspherical wave solutions [3,33]. For point sources distrib-uted over the transducer face as shown in Fig. 3, the pres-sure field at x in the fluid due to the point sourcesdistributed over y on the transducer face can be expressedas

pðxÞ ¼Z

SA

expðikf rÞ4pr

dSðyÞ ð61Þ

S. Banerjee et al. / Ultrasonics 46 (2007) 235–250 243

where the constant A is proportional to the source strength.From the Rayleigh–Sommerfield theory the above integralcan be expressed as [3,33]

pðxÞ ¼ � ixq2p

ZS

v3ðyÞexpðikf rÞ

rdSðyÞ ð62Þ

where v3(y) is the particle velocity on the transducer surfacealong the direction x3. For constant velocity of the trans-ducer surface (v3 = v0) Eq. (62) is simplified to

pðxÞ ¼ � ixqv0

2p

ZS

expðikfrÞr

dSðyÞ ð63Þ

2.14. Ultrasonic field in presence of a fluid–solid interface

Plane wave in a fluid is incident on a plane interfacebetween a fluid and a solid half-space as shown in Fig. 3.The incident P-wave produces both transmitted P and S-waves, and a reflected P-wave. The generation of S-wavesfrom the incident P-wave is called the mode conversion.This mode conversion makes computation of the ultrasonicfield inside the solid medium more difficult.

The pressure field in fluid, at point P, can be computedby adding the contributions of the direct incident ray (R1)and the reflected ray (R3). Acoustic wave speed and densityof the fluid are denoted by cf and qf, respectively, as shownin Fig. 3.

We are interested in computing the acoustic pressure atpoint P in the fluid. As shown in the figure, point P receivesa direct ray (R1) from point C and a ray (R3) reflected bythe interface at point T. Position vectors of points C, T

and P are denoted by y, z and x, respectively, as shownin the figure.

2.15. Pressure field computation in fluid at point P

Let vectors A and B represent CT and TP, respectivelyin Fig. 3; then,

A ¼ ðz1 � y1Þe1 þ ðz2 � y2Þe2 þ ðz3 � y3Þe3

B ¼ ðx1 � z1Þe1 þ ðx2 � z2Þe2 þ ðx3 � z3Þe3

ð64Þ

Note that the magnitudes of vectors A and B are R2 andR3, respectively.

R2 ¼ fðz1 � y1Þ2 þ ðz2 � y2Þ

2 þ ðz3 � y3Þ2g

12

R3 ¼ fðx1 � z1Þ2 þ ðx2 � z2Þ2 þ ðx3 � z3Þ2g12

ð65Þ

Unit vectors bA ¼ AR2; bB ¼ B

R3; Unit vector n normal to the

interface is given by n ¼001

8<:9=;: Note that for an inclined

interface n ¼n1

n2

n3

8<:9=;From the problem geometry it can

be clearly seen that

n� bA ¼ n� bB ð66Þn � bA ¼ �n � bB ð67ÞLet

bA ¼ a1

a2

a3

8><>:9>=>;; bB ¼ b1

b2

b3

8><>:9>=>; ð68Þ

Substituting the above unit vector expressions in Eq. (66)we get

Det

e1 e2 e3

n1 n2 n3

a1 a2 a3

264375 ¼ Det

e1 e2 e3

n1 n2 n3

b1 b2 b3

264375

or, in an alternate representation,

0 �n3 n2

n3 0 �n1

�n2 n1 0

264375 a1

a2

a3

8><>:9>=>; ¼

0 �n3 n2

n3 0 �n1

�n2 n1 0

264375 b1

b2

b3

8><>:9>=>;

or

0 �1 0

�1 0 0

0 0 0

264375 a1

a2

a3

8><>:9>=>; ¼

0 �1 0

�1 0 0

0 0 0

264375 b1

b2

b3

8><>:9>=>;

or

�a2

a1

0

8><>:9>=>; ¼

�b2

b1

0

8><>:9>=>;

or

� z2�y2

R2

z1�y1

R2

( )¼

z2�x2

R3

� z1�x1

R3

( )ð69Þ

Similarly from Eq. (67):

n1 n2 n3½ �a1

a2

a3

8><>:9>=>; ¼ � n1 n2 n3½ �

b1

b2

b3

8><>:9>=>;

or

0 0 1½ �a1

a2

a3

8><>:9>=>; ¼ � 0 0 1½ �

b1

b2

b3

8><>:9>=>;

or

a3 ¼ �b3 )z3 � y3

R2

¼ z3 � x3

R3

ð70Þ

Solving the above equations:

z1 ¼ y1ðx3�z3Þ�x1ðz3�y3Þx3�2z3þy3

z2 ¼ y2ðx3�z3Þ�x2ðz3�y3Þx3�2z3þy3

ð71Þ

Note that, if point C is on the x1x2 plane then y3 = 0, and forthe fluid, x3 is between 0 and z3; therefore, the denominator

244 S. Banerjee et al. / Ultrasonics 46 (2007) 235–250

of Eq. (71) should never become zero. After obtaining z1

and z2, from Eq. (71), the lengths R2 and R3 can be easilyobtained from Eq. (65). To evaluate R1 one does not needz1 and z2. It is simply equal to

R1 ¼ fðx1 � y1Þ2 þ ðx2 � y2Þ

2 þ ðx3 � y3Þ2g

12 ð72Þ

Then the pressure field at point P can be obtained from thefollowing equation:

pP ðxÞ ¼ �ixqf v0

2p

ZS

expðikfR1ÞR1

dS � ixqfv0

2p

�Z

S

R � expfikfðR2 þ R3ÞgR2 þ R3

dS ð73Þ

In Eq. (73) the first integral corresponds to the wavepath CP and the second integral corresponds to the wavepath CTP. Expression of the reflection coefficient R is givenby

R ¼

qs

qf

2k2

k2s� 1

� �2

þ 4k2gsbs

k4s

�� gs

gf

qs

qf

2k2

k2s� 1

� �2

þ 4k2gsbs

k4s

�þ gs

gf

ð74Þ

where

Fig. 4. The pressure distribution along the central axis of the transducer inwater (a) at 1 MHz and (b) 2.2 MHz frequencies, computed by the surfaceintegral technique and matrix inversion technique.

k ¼ kf sin h1 ¼ kp sin h2P ¼ ks sin h2S

gf ¼ kf cos h1 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2

f � k2q

gS ¼ kp cos h2P ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2

p � k2q

bs ¼ ks cos h2S ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2

s � k2q

kf ¼ xcf; kp ¼ x

cp; ks ¼ x

cs

In this case, the incident angle is h1, transmitted angle is h2S

and h2P for S-wave and P-wave, respectively. Fluid densityis qf and solid density is qs, acoustic wave speeds in the fluidis cf. Acoustic P-wave speed in the solid is cp and S-wavespeed is cs.

3. Results and discussion

MATLAB codes have been developed to model theultrasonic field near the fluid–solid interface using theDPSM formulation presented in the earlier section. To

Fig. 5. Ultrasonic field (normal stress along x-axis [s11] in aluminum andpressure in water) generated by a bounde beam of central frequency1 MHz striking the interface (a) at normal incidence and (b) at criticalangle (30.42�) of incidence.

S. Banerjee et al. / Ultrasonics 46 (2007) 235–250 245

present the numerical results an aluminum solid half-spaceis considered over the water half-space. For conveniencefrom now onwards the x1, x2 and x3 axes are denoted asx, y and z axes, respectively. However, definitions of stres-ses remain the same. The length of the interface is taken as40 mm. There is a requirement of the maximum spacingallowed between neighboring point sources, minimumnumber of point sources required and the distance of thepoint sources from the interface. These issues have beendiscussed in Refs. [24,25,33,34]. The point source distancesdepend on the wavelength of the ultrasonic signal [33]. Inthe surface integral based DPSM technique, along both x

and y axes, infinite dimensions of the interface are consid-ered, whereas in the matrix inversion based DPSM tech-nique finite dimension of the interface has beenconsidered, as mentioned above. The wave speed in waterand density are taken as 1.48 km/s and 1 g/cc, respectively.P-wave and S-wave velocity in aluminum are taken as6.5 km/s and 3.13 km/s, respectively. Density of aluminumis taken as 2.7 g/cc.

Ultrasonic waves are generated by an ultrasonic trans-ducer that is placed in water at different inclinations rela-tive to the interface. The diameter of the transducer is4 mm. The developed code is first checked for accuracyand reliability. This is done by computing the pressure field

Fig. 6. Field generated by a 2.2 MHz ultrasonic beam striking the interface(a) normal stress along x-axis (s11), (b) normal stress along z-axis (s33), and (

variation along the central axis of the transducer placed inthe fluid near the fluid–solid interface. The pressure varia-tions in the fluid along the z-axis are obtained by two tech-niques for both 1 MHz and 2.2 MHz transducers. It shouldbe noted here that the applicability of the method includeswide range of ultrasonic frequencies. The DPSM techniquehas been recently used for modeling high frequency(1 GHz) acoustic microscope [35].

The pressure field in fluid is computed by the DPSMmatrix inversion technique and surface integral basedtechnique, and compared in Fig. 4a and b, at two differ-ent frequencies. The results are in good agreement. There-fore, it can be concluded that the finite interfaceconsidered in this analysis can approximately model theinfinite interface geometry. The pressure field in waterhas alternate peaks and dips because of the constructiveand destructive interferences between the incident andreflected beams. Subsequent results are generated withthe DPSM technique with matrix inversion becausealthough it is computationally more demanding comparedto the surface integral technique it is also more flexibleand can be extended to the fluid–solid interface of anygeometry (curved interface with multiple curvatures forexample) and to the multi-layered cases, that are currentlyunder investigation.

at the critical angle (30.42�). Plots show the pressure field in water andc) vertical displacement along z-axis (u3) in aluminum.

246 S. Banerjee et al. / Ultrasonics 46 (2007) 235–250

The relative positions of the fluid–solid interface and theultrasonic transducer are shown in Fig. 1. The problemgeometry considered here has a finite size transducer(4 mm diameter). However, the interface length in the in-plane direction and its width in the out-of-plane directionare large, much greater than the transducer diameter. Thisthree dimensional problem is solved by the DPSM tech-nique and then the computed ultrasonic field is plottedalong the vertical plane of symmetry of the problem geom-etry. One hundred point sources are distributed behind thetransducer face and additional point sources are placedalong the interface as shown in Fig. 1. The question ishow many point sources should be used to model the inter-face that are extended to infinity in both in-plane and out-of-plane directions. At the plane of symmetry (or thecentral vertical plane) two lines of point sources near theinterface exist as shown in Fig. 1. First the ultrasonic fieldis computed along the central plane with this one plane(consisting of two lines) of point sources. Then two moreplanes of point sources are added on two sides of the cen-tral plane sources and the field is again computed at thecentral plane. This process of adding two planes of sourceson two sides of the central plane is continued until the com-

Fig. 7. Field generated by a 2.2 MHz ultrasonic beam striking the interface atalong x-axis (s11), (b) normal stress along z-axis (s33), and (c) vertical displac

puted field at the central plane is converged. Note thatadditional source planes on two sides of the central planeonly affect the number of point sources along the interface.For the transducer 100 point sources are placed behind thetransducer face from the very beginning. Interestingly, theresults are found to converge with only three planes ofsources consisting of a total of six lines of sources – threelines above and three lines below the interface. However,if one is interested in computing the ultrasonic field atanother plane which is not necessarily the plane of symme-try then more lines of sources at the interface might benecessary.

On each side of the interface 79 point sources are placedon the central plane. Sources are placed along the interfacein the illuminated region and also well beyond the illumi-nated region. A total of 237 sources are then necessaryon each side of the interface to model the interface withthree lines of point sources. Increasing the number ofsources to 395 on five lines did not significantly alter thecomputed ultrasonic field in the central plane. The numer-ical results presented below has been generated with 395point sources distributed over five lines below the interfaceand another 395 sources above the interface and 100

45.42� angle. Plots show the pressure field in water and (a) normal stressement along z-axis (u3) in aluminum.

S. Banerjee et al. / Ultrasonics 46 (2007) 235–250 247

sources behind the transducer face. The number of pointsources taken for the wave field computation is based onthe convergence criterion of the DPSM technique [33].The convergence of the problem solution has been alsotested by increasing the number of point sources in thein-plane direction and at the transducer face. When thespacing between two neighboring point sources is less thanthe half wave-length then the problem is found to converge.Further increase in the number of point sources did notchange the computed results significantly. For most ofthe results presented in this paper the distance betweentwo neighboring point sources has been kept at wave-length/p (approximately 2.8 mm, considering wave velocityin fluid). Thus the results presented in this paper are wellconverged.

At 2.2 MHz with the above number of point sources ittook approximately 14 s to solve the system of equationsin MATLAB in a Intel Pentium 4, 2.8 GHz processor.The computational time can be reduced significantly byconverting the programs in to C++ or FORTRAN whenthe programs can be run using direct compatible compilers.

In the following figures the grey scale bars are providedto give an idea about the magnitudes of the ultrasonic fieldsin the contour plots. Note that the grey scale bars are not

Fig. 8. Field generated by a 2.2 MHz ultrasonic beam striking the interface aalong x-axis (s11), (b) normal stress along z-axis (s33), and (c) vertical displac

identical in all figures. Fig. 5a shows the ultrasonic field(normal stress (s11) in aluminum and pressure in water)for normal incidence of the wave and Fig. 5b shows itfor the critical angle (30.42�) of incidence. Transducer fre-quency is 1 MHz for Fig. 5. Fig. 6a–c are generated with a2.2 MHz transducer for only critical angle of incidence. Itis well known that as the frequency of the signal increasesits wavelength decreases and the guided wave penetrationdepth decreases since it is proportional to the wavelength.Comparing Figs. 5b and 6a, one can see that the Rayleighwave penetration depth in the solid is relatively higher for1 MHz signal. The leaky waves in water are clearly visiblein Figs. 5b, 6a and b. It is interesting to note that even thenull region predicted by Bertoni and Tamir [36] in thereflected beam profile can be seen in the computed results.It should also be noted that the beam is more collimated at2.2 MHz.

Fig. 7a to c show the ultrasonic field in aluminum andwater at 2.2 MHz frequency, for 45.42� angle of incidence,which is 15� greater than the Rayleigh angle. It is wellknown that if the angle of incidence of the signal is greaterthan the critical angle, then most of the incident energy isreflected by the interface. Therefore, very small amountof energy should be transmitted inside the solid half-space.

t 15.42� angle Plots show the pressure field in water and (a) normal stressement along z-axis (u3) in aluminum.

248 S. Banerjee et al. / Ultrasonics 46 (2007) 235–250

This phenomenon is clearly visible in Fig. 7a–c. The pres-sure distribution in water and normal stresses s11 ands33 in aluminum are shown in Fig. 7a and b, respectivelyfor 2.2 MHz signal frequency. In Fig. 7a–c a very weakguided wave is observed although the incident angle isnot the critical angle. The guided wave is observed becausethe incident beam is not perfectly collimated. Therefore, asmall amount of energy still strikes the interface at the crit-ical angle.

Fig. 8a–c show the ultrasonic field in aluminum andwater at 2.2 MHz frequency, for 15.42� angle of incidence,which is 15� less than the critical angle. When the angle ofstrike is less than the critical angle then relatively moreenergy (in comparison to Figs. 6 and 7) is transmitted into the solid half-space as seen in Fig. 8a–c. Strong transmit-ted ultrasonic fields are clearly visible in Fig. 8. The wavespeed in aluminum is much higher than that in water.Therefore, when the ultrasonic beam hits the interface itis expected that, the transmitted field should propagate inaluminum with a higher transmission angle than the angleof incidence in water. This phenomenon can be clearly seenin Fig. 8a and b.

Different angles of inclination of the transducer withtwo different frequencies of excitation are considered.From the above results one can see that the wave reflection,transmission and generation of Leaky Rayleigh-waves canbe properly modeled by the DPSM technique.

4. Conclusion

DPSM technique is used to model the ultrasonic fieldgenerated by a finite size transducer near a fluid–solid inter-face. The field is computed inside both fluid and solidmedia. One major advantage of this technique is that itdoes not require any ray tracing or computation of reflec-tion and transmission coefficients at the interface. Thusextending this technique to multi-layered solid modelsshould be straightforward [37]. Several other advantagesof the DPSM technique like its capability of modeling crit-ical reflection phenomenon, etc. compared to other avail-able techniques such as the multi-Gaussian beammodeling technique has been discussed under Section 1.The computed results show how the finite size transducersinclined at the Rayleigh critical angle generate leaky Ray-leigh waves at the fluid–solid interface. To the best of ourknowledge these types of visual image of the critical reflec-tion phenomenon clearly showing the incident beam,reflected beam, the surface skimming leaky wave and thenull region has not been presented in the literature yet.

Acknowledgements

Financial supports from the National Science Founda-tion grants CMS-9901221 and OISE-0352680 are gratefullyacknowledged.

Appendix 1

Displacement Green’s function

Gijðx; yÞ ¼ 1

4pqx2

eikpr

rk2

pRiRj þ ð3RiRj � dijÞikp

r� 1

r2

� �� ��þ eiksr

rk2

s ðdij � RiRjÞ � ð3RiRj � dijÞiks

r� 1

r2

� �� ��where Ri ¼

xi � yi

rðA:1Þ

Let

ep ¼ eikpr

r; es ¼ eiksr

rand

rp ¼ ikp

r� 1

r2

� �; rs ¼ iks

r� 1

r2

� �ðA:2Þ

So we can write,

Gp11 ¼1

4pqx2½epðk2

pR21 þ ð3R2

1 � 1ÞrpÞ�;

Gs11 ¼1

4pqx2½esðk2

s ð1� R21Þ � ð3R2

1 � 1ÞrsÞ� ðA:3Þ

Gp12 ¼1

4pqx2½epðk2

pR1R2 þ ð3R1R2ÞrpÞ�;

Gs12 ¼1

4pqx2½esðk2

s ð�R1R2Þ � ð3R1R2ÞrsÞ� ðA:4Þ

Gp13 ¼1

4pqx2½epðk2

pR1R3 þ ð3R1R3ÞrpÞ�;

Gs13 ¼1

4pqx2½esðk2

s ð�R1R3Þ � ð3R1R3ÞrsÞ� ðA:5Þ

Gp22 ¼1

4pqx2½epðk2

pR22 þ ð3R2

2 � 1ÞrpÞ�;

Gs22 ¼1

4pqx2½esðk2

s ð1� R22Þ � ð3R2

2 � 1ÞrsÞ� ðA:6Þ

Gp21 ¼ Gp12; Gs21 ¼ Gs12 ðA:7Þ

Gp23 ¼1

4pqx2½epðk2

pR2R3 þ ð3R2R3ÞrpÞ�;

Gs23 ¼1

4pqx2½esðk2

s ð�R2R3Þ � ð3R2R3ÞrsÞ� ðA:8Þ

Gp31 ¼ Gp13; Gs31 ¼ Gs13 ðA:9Þ

Gp32 ¼1

4pqx2½epðk2

pR3R2 þ ð3R3R2ÞrpÞ�;

Gs32 ¼1

4pqx2½esðk2

s ð�R3R2Þ � ð3R3R2ÞrsÞ� ðA:10Þ

Gp33 ¼1

4pqx2½epðk2

pR23 þ ð3R2

3 � 1ÞrpÞ�;

Gs33 ¼1

4pqx2½esðk2

s ð1� R23Þ � ð3R2

3 � 1ÞrsÞ� ðA:11Þ

Gij ¼ Gpij þ Gsij ðA:12Þ

Differentiation of displacement Green’s function with respect

to x1, x2, x3

S. Banerjee et al. / Ultrasonics 46 (2007) 235–250 249

Let di the differentiation of the corresponding parameterw.r.t xi.

rpd1 ¼ 2R1

r3� ikpR1

r2

� �; rsd1 ¼ 2R1

r3� iksR1

r2

� �ðA:13Þ

rpd2 ¼ 2R2

r3� ikpR2

r2

� �; rsd2 ¼ 2R2

r3� iksR2

r2

� �ðA:14Þ

rpd3 ¼ 2R3

r3� ikpR3

r2

� �; rsd3 ¼ 2R3

r3� iksR3

r2

� �ðA:15Þ

again assuming eoij = RiRj and etij = 3RiRj

Note: Here index does not mean summation.

eoiidi¼�2R3i

rþ2Ri

r; eoijdi¼�2R2

i Rj

rþRj

r¼ eojidi ðA:16Þ

eoijdk¼�2RiRjRk

r; eoiidj¼�2RiRiRj

rðA:17Þ

etiidi¼�6R3i

rþ6Ri

r; etijdi¼�6R2

i Rj

rþ3Rj

r¼ etjidi ðA:18Þ

etijdk¼�6RiRjRk

r; etiidj¼�6RiRiRj

rðA:19Þ

Substituting the above expressions in the differentiation ofdisplacement Green’s function we getNote: Index does not mean summation and di representsdifferentiation of the corresponding parameters w.r.t xi.

Gpiidi ¼ 1

4pqx2½epðk2

peoiidiþ ð�1þ etiiÞrpdiþ rp � etiidiÞ�

þ Gpii ikpRiep � Riepr

� �ðA:20Þ

Gsiidi ¼ 1

4pqx2½esð�k2

s eoiidi� ð�1þ etiiÞrsdiþ rs � etiidiÞ�

þ Gsii iksRies� Riesr

� �ðA:21Þ

Gpiidj ¼ 1

4pqx2½epðk2

peoiidjþ ð�1þ etiiÞrpdjþ rp � etiidjÞ�

þ Gpii ikpRjep � Rjepr

� �ðA:22Þ

Gsiidj ¼ 1

4pqx2½esð�k2

s eoiidj� ð�1þ etiiÞrsdjþ rs � etiidjÞ�

þ Gsii iksRjes� Rjesr

� �ðA:23Þ

Gpijdi ¼ 1

4pqx2½epðk2

peoijdiþ ðetijÞrpdiþ rp � etijdiÞ�

þ Gpij ikpRiep � Riepr

� �ðA:24Þ

Gsijdi ¼ 1

4pqx2½esð�k2

s eoijdi� ðetijÞrsdiþ rs � etijdiÞ�

þ Gsij iksRies� Riesr

� �ðA:25Þ

Gpjidi ¼ 1

4pqx2½epðk2

peojidiþ ðetjiÞrpdiþ rp � etjidiÞ�

þ Gpji ikpRiep � Riepr

� �ðA:26Þ

Gsjidi ¼ 1

4pqx2½esð�k2

s eojidi� ðetjiÞrsdiþ rs � etjidiÞ�

þ Gsji iksRies� Riesr

� �ðA:27Þ

Gpijdk ¼ 1

4pqx2½epðk2

peoijdk þ ðetijÞrpdk þ rp � etijdkÞ�

þ Gpij ikpRkep � Rkepr

� �ðA:28Þ

Gsijdk ¼ 1

4pqx2½esð�k2

s eoijdk � ðetijÞrsdk þ rs � etijdkÞ�

þ Gsij iksRkes� Rkesr

� �ðA:29Þ

Giidi ¼ Gpiidiþ Gsiidi ðA:30ÞGiidj ¼ Gpiidjþ Gsiidj ðA:31ÞGijdi ¼ pijdiþ Gsijdi ðA:32ÞGjidi ¼ Gpjidiþ Gsjidi ðA:33ÞGijdk ¼ Gpijdk þ Gsijdk ðA:34Þ

Stresses at x due to a concentrated force at y can be writtenas

r133 ¼ ð2lþ kÞðG31d3Þ þ kðG11d1þ G21d2Þ

r233 ¼ ð2lþ kÞðG32d3Þ þ kðG12d1þ G22d2Þ ðA:35Þ

r333 ¼ ð2lþ kÞðG33d3Þ þ kðG13d1þ G23d2Þ

r111 ¼ ð2lþ kÞðG11d1Þ þ kðG21d2þ G31d3Þ

r211 ¼ ð2lþ kÞðG12d1Þ þ kðG22d2þ G32d3Þ ðA:36Þ

r311 ¼ ð2lþ kÞðG13d1Þ þ kðG23d2þ G33d3Þ

r131 ¼ lðG31d1þ G11d3Þ

r231 ¼ lðG32d1þ G12d3Þ ðA:37Þ

r331 ¼ lðG33d1þ G13d3Þ

r132 ¼ lðG31d2þ G21d3Þ

r232 ¼ lðG32d2þ G22d3Þ ðA:38Þ

r332 ¼ lðG33d2þ G23d3Þ

where rnij represents the stress rij at x due to the concen-

trated load acting along n (n = 1,2,3) at y.So we can write the expression for stress at any point

due to a three dimensional concentrated force acting inthe solid at another point. This expression is very usefulto calculate the total wave field at a point in solid, due toa group of concentrated force acting at the boundary orat the interfaces modeled by distributed point sourcemethod (DPSM).

sijðx; yÞ ¼ 1

4pðr1

ijP 1 þ r2ijP 2 þ r3

ijP 3Þ ¼ sijP

4pðA:39Þ

where

sij ¼ r1ij r2

ij r3ij

� and P ¼ P 1 P 2 P 3½ �T ðA:40Þ

250 S. Banerjee et al. / Ultrasonics 46 (2007) 235–250

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