modeling linear rayleigh wave sound fields generated by angle beam wedge transducers

Modeling linear Rayleigh wave sound fields generatedby angle beam wedge transducers Shuzeng Zhang,1Xiongbing Li,1,aHyunjo Jeong,2and Hongwei Hu3 1School of Traffic and Transportation Eng

Shuzeng Zhang, Xiongbing Li, Hyunjo Jeong, and Hongwei Hu

Citation: AIP Advances 7, 015005 (2017); doi: 10.1063/1.4972058

View online: http://dx.doi.org/10.1063/1.4972058

View Table of Contents: http://aip.scitation.org/toc/adv/7/1

Published by the American Institute of Physics

Modeling linear Rayleigh wave sound fields generated

by angle beam wedge transducers

Shuzeng Zhang,1Xiongbing Li,1,aHyunjo Jeong,2and Hongwei Hu3

1School of Traffic and Transportation Engineering, Central South University, Changsha,

Hunan 410075, China

2Division of Mechanical and Automotive Engineering, Wonkwang University, Iksan,

Jeonbuk 570-749, South Korea

3College of Automotive and Mechanical Engineering, Changsha University of Science

and Technology, Changsha, Hunan 410114, China

(Received 1 August 2016; accepted 28 November 2016; published online 4 January 2017)

In this study, the reciprocity theorem for elastodynamics is transformed into inte-gral representations, and the fundamental solutions of wave motion equations are obtained using Green’s function method that yields the integral expressions of sound beams of both bulk and Rayleigh waves In addition to this, a novel surface integral expression for propagating Rayleigh waves generated by angle beam wedge trans-ducers along the surface is developed Simulation results show that the magnitudes

of Rayleigh wave displacements predicted by this model are not dependent on the frequencies and sizes of transducers Moreover, they are more numerically stable than those obtained by the 3-D Rayleigh wave model This model is also applica-ble to calculation of Rayleigh wave beams under the wedge when sound sources are assumed to radiate waves in the forward direction Because the proposed model takes into account the actual calculated sound sources under the wedge, it can be applied

to Rayleigh wave transducers with different wedge geometries This work provides

an effective and general tool to calculate linear Rayleigh sound fields generated by

angle beam wedge transducers © 2017 Author(s) All article content, except where

otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license ( http://creativecommons.org/licenses/by/4.0/ ) [http://dx.doi.org/10.1063/1.4972058]

I INTRODUCTION

Rayleigh waves propagating on an elastic solid surface are commonly used in nondestructive testing and evaluation The energy of Rayleigh waves is concentrated in a thin layer, approximately one wavelength in depth, along the surface of the substrate; therefore, they can be used to test the surface cracks, hardness, and crystal structure, as well as thickness of the coatings and residual stress distribution, and others.1,2 Another advantage of the surface waves is that they are less affected by beam spreading than the bulk waves because they are confined to travel on the surface of the material and diverge only in one dimension rather than in two like in the case of the bulk ones Therefore, their energy can be concentrated in the main sound beam that makes them propagate a longer distance.3 Rayleigh waves can be generated by angle beam wedge transducers, comb transducers, inter-digital transducers, etc.47 Regarding an angle beam wedge transducer, a bulk P-wave transducer is placed on a wedge of a low speed material with a certain angle Because Rayleigh waves can be obtained efficiently and their energy is very high, angle beam wedge transducers are widely used in nondestructive evaluation with ultrasonic surface waves.8,9To consider the effects of beam splitting and distortion, reduction of the deviation between wave propagation and energy flow, and improve-ment the accuracy of Rayleigh wave detection, it is necessary to calculate accurate Rayleigh wave sound fields generated by angle beam wedge transducers

a Author to whom correspondence should be addressed Electronic mail: lixb ex@163.com

2158-3226/2017/7(1)/015005/13 7, 015005-1 © Author(s) 2017

Lord Rayleigh found the solutions of the equation of motion that represented traveling wave solutions which were confined primarily to a region near the surface of a semi-infinite space in 1887, since when Rayleigh waves have been widely studied A series of detailed works for calculation of the linear Rayleigh wave fields were provided by Achenbach and co-works,10–13in which, elastodynamic reciprocity approach and Fourier transform approach were developed to model the Rayleigh wave fields In their models, an ideal point load or line load is introduced as the sound source, so that such methods can not be directly used for modeling the Rayleigh wave fields generated by an angle beam wedge transducer An approach to solve this problem is to use a superposition of Green’s functions in which a P-wave transducer and a wedge are replaced by a line source with simple specific stress distri-butions on the surface.14–16Although this approach provides some basic characteristics of propagating Rayleigh waves, it does not take into account the waves generated in the transducer wedge, and thus, the simple stress distributions are not representative of the actual stresses present In addition, a line source used in the integral expression does not agree with the area sound sources under the wedge As a result, differences between the calculated Rayleigh sound distributions and real ones will be observed Recently, Schmerr et al.17have proposed a 3-D point source model for the surface beam Their method is based on the initial work by Aki and Richards,18in which, Rayleigh waves are assumed

to propagate in a laterally homogeneous medium and the magnitudes of two horizontal components

of Rayleigh waves are assumed to be dependent on the positions of the source and receiver.19This 3-D model was further developed by using the Fourier transform approach and simplified with a multi-Gaussian beam model.20,21 Although this 3-D model addresses all of the above mentioned inadequacies, it still has some limitations It has difficulties predicting the correct magnitudes of Rayleigh sound beams, because the integral relation over an area with 2-D Green’s function does not match with reciprocal theorem for 2-D wave motions and will bring confused results, and the Fourier transform approach provides quite large magnitude values for displacements (the results for magnitudes of displacements solved by Fourier transform approach should be those for displacement potentials, and one can find this problem through the detailed derivation of Sec 4 in the Ref.13) Therefore, significant different magnitudes of displacements can be obtained when wedge geometries and wave frequencies change while the initial displacements of the deriving transducer are fixed It has difficulties calculating Rayleigh sound beams under the wedge, which contributes to the generation of the second harmonic Rayleigh waves, and it is less appropriate for describing nonlinear propagation effects.22Although this model provides a useful technique to investigate the Rayleigh sound sources under the wedge, further research is still required

The main objective of this work is to develop a universal and accurate calculation method for modeling Rayleigh wave sound fields generated by angle beam wedge transducers This study is structured as follows Sec.IIbriefly illustrates the generation process of Rayleigh waves by an angle beam wedge transducer In the same section, we derive the expressions for propagating Rayleigh waves and analyze sound sources that generate Rayleigh waves Sec.IIIpresents the application of elastodynamic reciprocity to the angle beam wedge transducers for the purposes of modeling Rayleigh sound beams Additionally, we develop a surface integral expression for Rayleigh waves, in which the surface sound sources agree with the actual calculated area sources under the wedge Finally, Sec.IVprovides the simulation results of Rayleigh sound fields and discusses specific advantages of the proposed method

II RAYLEIGH WAVE GENERATION BY ANGLE BEAM WEDGE TRANSDUCERS

As shown in Fig.1, a contact bulk transducer radiates a P-wave into a wedge, and Rayleigh waves are generated on the surface of a specimen, which is an isotropic elastic half-space solid when the P-wave propagates to the specimen through the interface with a chosen incident angle θ The

origin of the coordinate system (x1y1z1) is in the center of transducer, and the coordinates (x2y2z2) are located at the intersection of the transducer center axis and the specimen surface Furthermore, the

axes z1 and z2are normal to the transducer and the plane of the stress-free surface of the specimen, respectively

Here, we focus exclusively on the generation of Rayleigh waves in the x-z plane When the

bulk-wave transducer radiates the P-bulk-wave into the wedge and this bulk-wave incidents the smooth interface

FIG 1 Schematic diagram of an angle beam wedge transducer for generating Rayleigh waves.

between the wedge and the specimen, both generated P- and S-waves will be transmitted Thus, we can represent the incident P-wave in a displacement potential form as

ϕinc= Aiexp(ikp1(x sin θp1+ z cos θp1) − iωt), (1) and the transmitted P- and S-wave as

φtp= Atexpfikp2(x sin θp2+ z cos θp2) − iωtg, (2a)

ψts= Btexp [iks2(x sin θs2+ z cos θs2) − iωt] , (2b)

where Ai, Atand Btare the amplitudes of potentials for the incident wave and the transmitted

P-and S-waves, respectively, k = ω/c is the wave number, θp1is the incident angle, θp2and θs2are the transmitted angles for the transmitted P- and S-wave, respectively Note that we use an assumption that the contact transducer radiates only P-wave, and the P- and S-waves reflected back to wedge are not listed here

When the incident angle is larger than the second critical angle, both transmitted waves are inhomogeneous At a certain incident angle, these P- and S-waves start mixing and propagating with the same apparent wave speed on the surface of a semi-infinite space In this case, the waves are well-known as Rayleigh waves And the incident angle can be calculated using Snell’s law when the speeds of incident wave and Rayleigh wave are known Note that Rayleigh wave can be effectively generated in a small angle range near this calculated one Below, we introduce a Rayleigh wave number and demonstrate Snell’s law as

kr= kp1sin θp1= kp2sin θp2= ks2sin θs2, (3)

where kr= ω/cr, and cris the speed of Rayleigh wave Then, the inhomogeneous P- and S- waves

can be written in terms of krand cras

φtp= Atexp









−z |kr| * ,

1 − c

2 r

c2p2+

-1/2







exp [ikr(x − crt)] , (4a)

ψts= Btexp









−z |kr| * ,

1 − c

2 r

c2 s2 +

-1/2







exp [ikr(x − crt)] (4b)

It can be seen that these waves propagate mainly along the surface direction and decay

expo-nentially in the depth direction They can be treated as the z- and x-components of Rayleigh waves,

respectively We introduce α and β to replace the decay parts as

α = * ,

1 − c

2 r

c2 p2 +

-1/2 , β= * ,

1 − c

2 r

c2 s2 +

-1/2

Then, we can obtain the expression for displacement and stresses through the wave motion function from Eqs (4a,4b), as follows

u = [ik A exp(−αk z) − k βB exp(− βk z)] exp [ik (x − c t)] , (6a)

uz= [−αkrAtexp(−αkrz) − ikrBtexp(− βkrz)] exp [ikr(x − crt)] , (6b)

τzz= µf

(2kr2−k2s)Atexp(−αkrz) + 2ik2

rβBtexp(− βkrz)gexp [ikr(x − crt)] , (6c)

τxz= µf

−2ikr2αAtexp(−αkrz) + (k2

r + β2kr2)Btexp(− βkrz)gexp [ikr(x − crt)] , (6d) where µ is the Lam´e constant

To determine an explicit value of the Rayleigh wave speed, the stress-free boundary condition

at the interface y= 0 of the specimen must be satisfied with the following expressions:

τzz=

2kr2−ks22At+ 2ik2

τxz= −2ik2

rαAt+

kr2β2+ k2

r



Bt= 0 (7b)

Then, we can obtain the relationship between Atand Btfrom Eq (7b) as

At= −β2i β2

It can also be seen that this stress-free boundary condition requires that

*

,

2 − c

2 r

c2 s2 +

-2

−4 * ,

1 − c

2 r

c2 p2 +

-1/2

*

,

1 − c

2 r

c2 s2 +

-1/2

which is the equation for the phase velocity of Rayleigh waves

Next, we apply the Eqs (6c,6d) again to consider the boundary condition under the wedge Because the wedge and the specimen are contacted smoothly through a thin liquid couplant, the boundary condition under the wedge satisfies the following expressions:

τzz=

2kr2−ks22At+ 2ik2

τxz= −2ik2

rαAt+

kr2β2+ k2

r



In Eq (10a) that the pressures, p, under the wedge are the same as those acting on the specimen

below, thus, they serve as the sources of Rayleigh waves.17 These pressures are generated by the contact bulk transducer and can be accurately calculated with the integral method

Furthermore, the relationship between the amplitudes of the potential and the displacement amplitudes can be used to simplify the expressions of Rayleigh wave displacements in Eqs (6a,6b)

We introduce ur(x, y) as the propagation Rayleigh wave beam on the surface of the specimen, so that

we can describe the displacement amplitudes of Rayleigh waves in the three-dimensional coordinates as

ux(x, y, z, t) = ur(x, y) [− 2i β

β2+ 1exp(−αkrz) + i β exp(−βkrz)] exp [ikr(x − crt)] , (11a)

uz(x, y, z, t) = ur(x, y) [ 2α β

β2+ 1exp(−αkrz) − exp(− βkrz)] exp [ikr(x − crt)] (11b)

In these expressions, the first terms in the brackets represent the Rayleigh sound beams, and the second and third terms are the polarization vectors and the phase terms, respectively Note that

the y-components of Rayleigh waves are neglected because they are of a higher order than the

x-components These equations are similar to those for describing Rayleigh waves in the Ref.4because the stress-free boundary condition τxz= 0 is introduced to obtain the ratio of z- and x-components.

However, in other research, the stress-free boundary condition τzz= 0 is used which will bring different solutions.12,13Which boundary condition should be used is out of discussion in this study

Now, we have derived the expressions for propagating Rayleigh waves and explained sound sources for generating Rayleigh waves In the next section, we will show how to obtain the Rayleigh

sound sources under the wedge and model Rayleigh wave sound beams ur(x, y) on the surface of the

specimen based on the reciprocity theorem

III INTEGRAL EXPRESSIONS FOR RAYLEIGH WAVE SOUND FIELDS

We apply the reciprocity theorem to develop the integral representation theorem, and derive fundamental solutions of the wave motion equations using Green’s function method This section combines the fundamental solutions with the integral representations to obtain integral expressions for sound fields of both bulk and Rayleigh waves

First, we show demonstrate how to obtain the P-wave sound fields radiated by the contact transducer which will then be used as the Rayleigh sound source in the subsequent part For a region

V with the boundary S, the reciprocal theorem can be written as8



V (f k1u2k−f k2u1k )dV=



S

(τkl2n k u1l −τ1

where τ1

kland τ2

kl are the stresses, f1

k and f2

k are the body forces, u1

k and u2

kare the displacements All

these parameters come from the two different equations of motion n kis the normal vector perpen-dicular to the surface of the medium The reciprocal theorem is a formal mathematical relationship between the fields of two different solutions, but it can be transformed into a very useful integral equation if we choose a proper solution of the reciprocal theorem from Eq (12) can be transformed into very useful integral equation To model P-wave sound fields radiated by a piston transducer, we

apply the theorem to the half-space V (z ≥ 0), shown in Fig.2, and assume the radiated waves satisfy

the Sommerfeld radiation condition and the displacement at a point x1(x1, y1, z1) can be written as

u1(x1, ω)=



ST

"

C klij n(x1)G(x1, y1, ω)∂u0(y1, ω)

∂n(y1) −C klij n(x1)u0(y1, ω)

∂G(y1, ω)

∂n(y1)

#

dST(y1), (13)

where u0is the initial displacement of a particle in the transducer sources, and STis the transducer face This representation can be simplified as

u1(x1, ω)=



ST

"

2C klij n(x1)G(x1, y1, ω)∂u0(y1, ω)

∂n(y1)

#

dST(y1) (14)

The fundamental solution for an elastic solid in Eq (14) is the solution of Navier’s equations for a delta function body force term However, if we consider only P-wave radiated by the contact transducer and propagating in the wedge, Navier’s equations can be simplified to have the same form

of the wave equations as that in fluid Hence, instead of Navier’s equations, a 3-D wave motion equation is used to simplify the solution as follows:

*

,

∂2u

∂x2 1

+∂∂y2u2 1

+∂∂z2u2 1 +

-− 1

c2

∂2u

∂t2 = −f (t) exp(iωt). (15) The solution, so-called Green’s function, can be obtained with Green’s function method23

G(x1, y1, ω)=exp(ikR1)

4πR1

where R1= x1− y1 is the distance from the sound source position to the target point Note that the solutions treat longitudinal wave propagation in isotropic elastic solids as being similar to that in liquids; that is, it ignores any mode conversion in the solid In addition, this assumption enables us

FIG 2 Geometry for calculating bulk wave sound fields generated by area sources in a three-dimensional coordinate system.

to convert between longitudinal particle velocities and displacements Then, we substitute Green’s function from Eq (16) into the integral representation in Eq (14) and obtain to yield the bulk wave displacement fields in the wedge as

u1(x1, ω)=v0(y1)

2iπcp1



S T

exp(ikp1R1)

R1 dST(y1), (17)

where v0(y1)= p0(y1) ρ1cp1= iωu0(y1) is the initial particle velocity in the sound source It has been demonstrated that when only P-wave radiated by a contact transducer is considered in a solid, this expression is exactly the form same as one for a piston transducer radiating into a fluid, as shown in the Ref.9

Next, we apply the reciprocity theorem to modeling the Rayleigh wave sound beams As we mentioned in Sec.II, the energy of Rayleigh waves is concentrated on the surface of the substrate and decays exponentially with depth Additionally, Rayleigh waves on a free surface can be expressed

in a general form with a 2-D wave equation.10Thus, it is reasonable to describe the Rayleigh sound

beams in the x-y plane as shown in Fig.3 We write the reciprocal theorem in an arbitrary surface

area S of an elastic isotropic solid with a boundary L as



S (f k1u2k−f k2u1k )dS=



L

(τkl2n k u1l −τ1

where the parameters are the same as in Eq (12), except that the volume integral and the surface integral in Eq (12) have been replaced by the surface integral and the line integral, respectively

Applying this theorem into the half-space S(x ≥ 0) (Fig.3) and assuming that Rayleigh waves radiated from the sound source satisfy the Sommerfeld radiation conditions, we can obtain the relationship

between the sound source and the displacements of Rayleigh waves at the point x2(x2, y2, z2) on the surface as

ur(x2, ω)=



LS

"

2C klij n(x2)G(x2, y2, ω)∂u0

1(y2, ω)

∂n(y2)

#

dLS(y2), (19)

where u10 is the displacement of the source required to generate Rayleigh waves, LSis the length of the source In Eq (19), G becomes the fundamental solution to the 2-D wave motion equation Here,

the 2-D wave motion equation is given as

*

,

∂2ur

∂x2 2

+∂2ur

∂y2 2 +

-− 1

c2 r

∂2ur

and its Green’s function solution is23

G(x2, y2, ω)=−i

4

r

2i

krπR2 exp(ikrR2) (21) Similarly, the combination of the integral representation and Green’s function yields the Rayleigh wave sound beams expression as

ur(x2, ω)=−2

cr



LS

v10(y2)−i 4

r

2i

krπR2exp(ikrR2)dLS, (22)

FIG 3 Geometry for calculating Rayleigh wave sound field generated by line sources in a two-dimensional coordinate system.

where R2= x2− y2, v10(y2) is the particle velocity of the Rayleigh sound source.

It has been observed that the integration of the line sources did not match the actual calculated area sound sources when we modelled sound fields of angle beam wedge transducers As shown in Fig.4, we introduced an algebra method to solve this problem in this study

In Fig 4, the active region of the Rayleigh sound sources is in the so-called ‘footprint’ area which can be obtained using the exact geometry optics, when the sound beam in the wedge radiated

by the contact transducer is assumed well collimating.6 Firstly, we divide the area sound sources into multiple banding sources which are narrow enough to be treated as line sources Then, the area sound sources are replaced by these line sources and the line integral formula from Eq (22) can be extended to the following form to express the surface integral as

ur(x2, ω)= C



Ly

v10(y2)G(x2, y2)dLy= C

i

X

1

v1i0(y2i )G(x2, y2i )∆y i

= C

Lx



Ly

v10(y2)G(x2, y2)dLy×Lx= C

Lx

i

X

1

v10(y2i )G(x2, y2i )∆y i (∆x1+ · · · + ∆x j) (23)

= C

Lx

i

X

1

j

X

1

v10(y2ij )G(x2, y2ij )∆y i∆x j= C

Lx



S

v10(y2)G(x2, y2)dSF,

where C = −2/cr, and Lxis the equivalent length of the footprint area source SF, which is introduced

to calculate the sound beams out of the footprint and given as

where a is the radius of the contact transducer Note that the equivalent length Lxis introduced when the footprint is completely included underneath the wedge The cases in which the wedges partially cover the footprints will be discussed later

Before we obtain the final integral expression for Rayleigh waves, we had to address the problem with the calculated P-wave sound pressures that could not be employed directly as the Rayleigh sound sources The reflection and transmission coefficients for Rayleigh wave scattered by a wedge and for love wave propagating in a welded-contacted solid-solid interface have been researched.24–26 The continuous boundary condition and conservation of energy should be satisfied to obtain these coefficients Fortunately, when Rayleigh wave is generated by an obliquely incident longitudinal wave, a transmission coefficient is developed to make sure that the pressures in the wedge are the same as those acting on the specimen from below at the interface.17 The transmission coefficient comes from a high frequency approximation for the reflection of a point source by calculating the pressure at the interface from the incident and reflected waves in the wedge and is given as17

T=ρ1cp1

ρ2cr2

*

,

Tp;pcs2

cp2*

,

c2p2

c2 s2

−2

c2p2

c2 r2 +

-−2iTs;pcs2

cr2

v t

c2 s2

c2 r2

−1+/

FIG 4 Geometry for calculating Rayleigh wave sound field generated by area sources in a two-dimensional coordinate system.

where (Tp;p, Ts;p) are ordinary plane wave transmission coefficients for the P- and S-waves, respec-tively, for two elastic solids in smooth contact Note that a velocity-based transmission coefficient is introduced here to apply it directly to our model

Combining Eq (17) and Eq (25) yields the Rayleigh sound sources expression Then, we place the sources expression into Eq (23) and construct the final expression for the propagating Rayleigh wave beams outside of the region underneath the wedge as

ur(x, y)= 1

Lx

kp1v0(y1)T 2πcr

exp(iπ/4)

√

2krπ



SF



ST

exp(ikp1R1)

R1

exp(ikrR2)

√

R2 dSTdSF (26)

One can find that Eq (26) is similar to Eq (11) in the Ref.17, because the Rayleigh wave fields are both expressed as integrals over the transducer and footprint surface But the major difference is that

Eq (26) can still be effective when the footprint size changes, even the footprint surface is replaced

by a line In addition, the polarization vector terms are separated from this integral expression, and the Rayleigh sound beams are expressed in a simple form similar to that in Ref.4, therefore, Eq (26)

is very computationally efficient

IV SIMULATIONS AND DISCUSSIONS

A Efficiency of the surface integral expression

As evident from the Eq (26), Rayleigh sound beam distributions depend on two surface integrals: one is the integral over the contact transducer surface which is fixed, and another is over the footprint with its size determined by the wedge’s shape Note that in angle beam wedge transducers used to generate Rayleigh waves, there is usually no material in the wedge for acoustic damping that typically ensures that the energy exists only in the footprint.27 Consequently, it is necessary to evaluate the efficiency of the proposed surface integral expression for Rayleigh sound beams and the choice of

the equivalent length Lx

Fig.5shows the sound energy distributions under the wedge where the exact footprint is located

in the ellipse line and the equivalent length is the distance between the two dashed lines The radius

of the transducer is 6.35 mm and its center frequency is 5 MHz The P-wave speed and the incident angle for the Lucite wedge are 2700 m/s and 71.63 degree, respectively The P-wave, S-wave and the corresponding Rayleigh wave speeds in the aluminum sample are 6250 m/s, 3080 m/s and 2845 m/s, respectively The distance from the center of the transducer to the center of the footprint in the sample

is 25 mm We take the initial displacement magnitude in the transducer as unity Although we can see that the bulk of the energy is concentrated in the footprint, there are active regions outside of this

area because of the beam spreading To explain the efficiency of the selected Lxand the integral area,

we calculate the Rayleigh sound beams using different sizes of integral areas and compare them with those obtained from the exact integral footprint The simulation results, shown in Fig.6, have only z-components of the particle displacements

FIG 5 Schematic image of the sound pressure in footprint under the wedge The region in the ellipse line is the footprint

corresponding to the circle transducer, the distance between two vertical lines is the equivalent length L of the footprint.

FIG 6 The z-components of the particle displacements along (a) x-axis, and (b) y-axis, z = 0.2 m, calculated with different

integral areas.

Fig.6(a)compares the normalized z-components of the particle displacements along the

prop-agating distance using different sound source areas Note that the absolute values of displacements can be calculated through Eq (26), but here for comparison, these values are normalized by selecting the maximum values in each figure as 1 In this plot, the displacements are not given in the region near the original point It can be observed that the particle displacements agree well with each other

if the propagating distance is large with an infinite integral area, a greater integral area (1.2Lx×3a) and the theoretical footprint when the equivalent Lx from Eq (24) is introduced It shows that the

choice of Lxis reasonable and the active sound sources outside the main footprint have little effect

on the Rayleigh sound field amplitudes at a large propagating distance However, they show slightly different behaviors in the region close to the wedge, as we found that the sound sources outside

of the main footprint made these values larger Fig 6(b)shows the results along the y-axis at the propagation distance of 200 mm We also found that the integral sound source areas slightly affected the distributions of the particle displacements in the transverse axis direction Note that because the Rayleigh sound sources, shown in Fig.5, cannot be exactly expressed using a line source, the surface integral expression with calculated sources theoretically might provide more accurate Rayleigh wave distributions

B Rayleigh sound beams with different transducers

Three different transducers used in this study have 6.35 mm radius circular element and 5 MHz center frequency (Transducer 1), 6.35 mm and 2.5 MHz (Transducer 2), and 8 mm and 5 MHz (Transducer 3), respectively The other parameters are the same as those used in Sec.Aexcept that

the equivalent length Lxshould be recalculated if the transducer size changes The region underneath

the wedge is assumed large enough to cover the footprint The x- and z-components of the particle

displacements of Rayleigh waves on the x-axis are calculated and shown in Fig.7(a) The normaliza-tion method is the same as that mennormaliza-tioned above We found that the maximum displacements were

FIG 7 Magnitudes of the particle displacements with different transducers calculated using: (a) the proposed model, and (b) the 3-D model.