Modeling linear Rayleigh wave sound fields generated by angle beam wedge transducers 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 AIP ADVANCES 7, 015005 (2017) Modeling linear Rayleigh wave sound fields generated by angle beam wedge transducers Shuzeng Zhang,1 Xiongbing Li,1,a Hyunjo Jeong,2 and Hongwei Hu3 School of Traffic and Transportation Engineering, Central South University, Changsha, Hunan 410075, China Division of Mechanical and Automotive Engineering, Wonkwang University, Iksan, Jeonbuk 570-749, South Korea College of Automotive and Mechanical Engineering, Changsha University of Science and Technology, Changsha, Hunan 410114, China (Received August 2016; accepted 28 November 2016; published online January 2017) In this study, the reciprocity theorem for elastodynamics is transformed into integral 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 transducers 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 applicable 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, interdigital transducers, etc.4–7 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.8,9 To consider the effects of beam splitting and distortion, reduction of the deviation between wave propagation and energy flow, and improvement 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 015005-2 Zhang et al AIP Advances 7, 015005 (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,10–13 in 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 distributions on the surface.14–16 Although 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.17 have proposed a 3-D point source model for the surface beam Their method is based on the initial work by Aki and Richards,18 in 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.19 This 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 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.22 Although 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 II briefly 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 III presents 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 IV provides 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 (x1 y1 z1 ) is in the center of transducer, and the coordinates (x2 y2 z2 ) are located at the intersection of the transducer center axis and the specimen surface Furthermore, the axes z1 and z2 are 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 bulkwave transducer radiates the P-wave into the wedge and this wave incidents the smooth interface 015005-3 Zhang et al AIP Advances 7, 015005 (2017) FIG 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 = Ai exp(ikp1 (x sin θ p1 + z cos θ p1 ) − iωt), (1) and the transmitted P- and S-wave as φtp = At exp ikp2 (x sin θ p2 + z cos θ p2 ) − iωt , (2a) ψts = Bt exp [iks2 (x sin θ s2 + z cos θ s2 ) − iωt] , (2b) where Ai , At and Bt are the amplitudes of potentials for the incident P-wave and the transmitted Pand S-waves, respectively, k = ω/c is the wave number, θ p1 is the incident angle, θ p2 and θ s2 are 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 = kp1 sin θ p1 = kp2 sin θ p2 = ks2 sin θ s2 , (3) where kr = ω/cr , and cr is the speed of Rayleigh wave Then, the inhomogeneous P- and S- waves can be written in terms of kr and cr as 1/2 cr2 exp [ikr (x − cr t)] , (4a) φtp = At exp −z |kr | − cp2 1/2 cr2 exp [ikr (x − cr t)] ψts = Bt exp −z |kr | − (4b) cs2 It can be seen that these waves propagate mainly along the surface direction and decay exponentially 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 1/ 1/ c2 c2 α = − 2r , β = − 2r (5) cp2 cs2 Then, we can obtain the expression for displacement and stresses through the wave motion function from Eqs (4a, 4b), as follows ux = [ikr At exp(−αkr z) − kr βBt exp(− βkr z)] exp [ikr (x − cr t)] , (6a) 015005-4 Zhang et al AIP Advances 7, 015005 (2017) uz = [−αkr At exp(−αkr z) − ikr Bt exp(− βkr z)] exp [ikr (x − cr t)] , (6b) τzz = µ (2kr2 − ks2 )At exp(−αkr z) + 2ikr2 βBt exp(− βkr z) exp [ikr (x − cr t)] , (6c) τxz = µ −2ikr2 αAt exp(−αkr z) + (kr2 + β kr2 )Bt exp(− βkr z) exp [ikr (x − cr t)] , (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 = of the specimen must be satisfied with the following expressions: τzz = 2kr2 − ks2 At + 2ikr2 βBt = 0, (7a) τxz = −2ikr2 αAt + kr2 β + kr2 Bt = (7b) Then, we can obtain the relationship between At and Bt from Eq (7b) as At = − 2i β Bt β2 + (8) It can also be seen that this stress-free boundary condition requires that 2− cr2 cs2 −4 1− cr2 cp2 1/ 1− cr2 cs2 1/ = 0, (9) 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 − ks2 At + 2ikr2 βBt = −p, (10a) τxz = −2ikr2 αAt + kr2 β + kr2 Bt = (10b) 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 2i β ux (x, y, z, t) = ur (x, y) [− exp(−αkr z) + i β exp(− βkr z)] exp [ikr (x − cr t)] , (11a) β +1 uz (x, y, z, t) = ur (x, y) [ 2α β exp(−αkr z) − exp(− βkr z)] exp [ikr (x − cr t)] β2 + (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 xcomponents These equations are similar to those for describing Rayleigh waves in the Ref because the stress-free boundary condition τxz = is introduced to obtain the ratio of z- and x-components However, in other research, the stress-free boundary condition τzz = is used which will bring different solutions.12,13 Which 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 015005-5 Zhang et al AIP Advances 7, 015005 (2017) 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 (fk1 uk2 − fk2 uk1 )dV = S (τkl2 nk ul1 − τkl1 nk ul2 )dS, (12) where τkl1 and τkl2 are the stresses, fk1 and fk2 are the body forces, uk1 and uk2 are the displacements All these parameters come from the two different equations of motion nk is the normal vector perpendicular 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 Cklij n(x1 )G(x1 , y1 , ω) u1 (x1 , ω) = ST ∂u0 (y1 , ω) ∂G(y1 , ω) − Cklij n(x1 )u0 (y1 , ω) dST (y1 ), ∂n(y1 ) ∂n(y1 ) (13) where u0 is the initial displacement of a particle in the transducer sources, and ST is the transducer face This representation can be simplified as u1 (x1 , ω) = 2Cklij n(x1 )G(x1 , y1 , ω) ST ∂u0 (y1 , ω) dST (y1 ) ∂n(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 ∂2u ∂2u ∂2u + + − 2 = −f (t) exp(iωt) ∂x1 ∂y1 ∂z1 cp ∂t (15) The solution, so-called Green’s function, can be obtained with Green’s function method23 G(x1 , y1 , ω) = exp(ikR1 ) , 4πR1 (16) 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 Geometry for calculating bulk wave sound fields generated by area sources in a three-dimensional coordinate system 015005-6 Zhang et al AIP Advances 7, 015005 (2017) 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 ST exp(ikp1 R1 ) dST (y1 ), R1 (17) where v0 (y1 ) = p0 (y1 ) ρ1 cp1 = 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 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.10 Thus, it is reasonable to describe the Rayleigh sound beams in the x-y plane as shown in Fig We write the reciprocal theorem in an arbitrary surface area S of an elastic isotropic solid with a boundary L as S (fk1 uk2 − fk2 uk1 )dS = L (τkl2 nk ul1 − τkl1 nk ul2 )dL, (18) 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 ∂u (y2 , ω) dLS (y2 ), (19) ur (x2 , ω) = 2Cklij n(x2 )G(x2 , y2 , ω) ∂n(y2 ) LS where u1 is the displacement of the source required to generate Rayleigh waves, LS is 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 ∂ ur ∂ ur ∂ ur + − 2 = −f (t) exp(iωt), 2 ∂x2 ∂y2 cr ∂t (20) and its Green’s function solution is23 G(x2 , y2 , ω) = −i 2i exp(ikr R2 ) kr πR2 (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 v1 (y2 ) −i 2i exp(ikr R2 )dLS , kr πR2 (22) FIG Geometry for calculating Rayleigh wave sound field generated by line sources in a two-dimensional coordinate system 015005-7 Zhang et al AIP Advances 7, 015005 (2017) where R2 = x2 − y2 , v1 (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 i ur (x2 , ω) = C Ly = = C Lx C Lx v1 (y2 )G(x2 , y2 )dLy = C Ly i v1i (y2i )G(x2 , y2i )∆yi v1 (y2 )G(x2 , y2 )dLy × Lx = C Lx j v1 (y2ij )G(x2 , y2ij )∆yi ∆xj = 1 i v1 (y2i )G(x2 , y2i )∆yi (∆x1 + · · · + ∆xj ) (23) C Lx S v1 (y2 )G(x2 , y2 )dSF , where C = −2/cr , and Lx is the equivalent length of the footprint area source SF , which is introduced to calculate the sound beams out of the footprint and given as Lx = 2a/ cos θ p1 , (24) where a is the radius of the contact transducer Note that the equivalent length Lx is 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= 2 cp2 ρ1 cp1 p;p cs2 cp2 cs2 T − − 2iT s;p 2 ρ2 cr2 cp2 cs2 cr2 cr2 cs2 cr2 −1 , (25) FIG Geometry for calculating Rayleigh wave sound field generated by area sources in a two-dimensional coordinate system 015005-8 Zhang et al AIP Advances 7, 015005 (2017) where (T p;p , T s;p ) are ordinary plane wave transmission coefficients for the P- and S-waves, respectively, 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) = kp1 v0 (y1 )T exp(iπ/4) √ Lx 2πcr 2kr π SF ST exp(ikp1 R1 ) exp(ikr R2 ) dST dSF √ R1 R2 (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 shows 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 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 Lx and 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 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 x of the footprint 015005-9 Zhang et al AIP Advances 7, 015005 (2017) FIG 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 propagating 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 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 Lx is 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 MHz center frequency (Transducer 1), 6.35 mm and 2.5 MHz (Transducer 2), and mm and MHz (Transducer 3), respectively The other parameters are the same as those used in Sec A except that the equivalent length Lx should 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 normalization method is the same as that mentioned above We found that the maximum displacements were FIG Magnitudes of the particle displacements with different transducers calculated using: (a) the proposed model, and (b) the 3-D model 015005-10 Zhang et al AIP Advances 7, 015005 (2017) approximately the same Similar to the bulk wave sound field generated by a piston transducer, with a higher frequency and a larger size, Rayleigh sound waves can travel a longer distance with a low decrement For comparison, the normalized displacements of the x- and z-components calculated using the 3-D model are shown in Fig 7(b) As we have mentioned above, the calculated magnitudes of displacements may be the displacement potentials and the direct surface integral over the footprint may cause quite different results, thus, the normalized results obtained from 3-D model are used for comparison In general, these beam behaviors predicted by different models agree with each other Note that each component of Rayleigh waves have to be calculated separately in the 3-D model; however, in our proposed model, once the Rayleigh sound beam distributions have been obtained, the x- and z-components of waves are obtained with Eqs (11a, 11b) In this way we can avoid a more complicated calculation process that requires obtaining much lower y-components of Rayleigh waves, and use a much simpler approach for modeling Rayleigh sound fields propagating in isotropic and homogeneous solids In addition to this, we found that the maximum magnitudes of displacements predicted by the 3-D model were dependent on the frequencies and the sizes of transducers, which are different from the results obtained by our method Basically, the maximum velocities or displacements are dependent on the initial values of the sound source This is always true for the bulk waves of the contact or immersion transducers,8 and Rayleigh waves of the comb transducers.9 As shown in Fig 7(a), it is reasonable to predict more numerically stable magnitudes of velocities or displacements, based on the results obtained with our model, even if a wedge is placed between the transducer and the specimen C Rayleigh sound beams under the wedge Following the Rayleigh waves generation process, we assume that those Rayleigh sound sources radiate waves only in the forward direction Thus, these Rayleigh wave beams underneath the wedge can be calculated using the sound sources ahead which contribute to the generation of Rayleigh waves Note that when the Rayleigh sound beams are calculated, the equivalent length Lx is selected for the new values, as illustrated in Fig Fig compares the Rayleigh sound beams under the wedge obtained with two different methods The normalized results are shown in these figures The parameters used in this section are the same as those in Sec A Fig 8(a) demonstrates the results calculated using an equation similar to Eq (26) but based on the above mentioned method, while Fig 8(b) shows the results obtained directly from the 3D model Note that only normalized sound beam distributions of thez-components are illustrated The obtained differences are significant More specifically, there are no Rayleigh sound field distributions in the opposite propagation direction in Fig 8(a), while symmetric Rayleigh sound field distributions exist in Fig 8(b) if we neglect the radiating directions of the sound sources, which is not true in the actual experiments In addition to this, the sound fields under the wedge also show significant differences Note that the wave distribution results under the wedge may be same if we consider the propagation direction in both models, but the stationary phase method in that 3-D model can not be employed FIG Comparison of the Rayleigh sound field distributions under the wedge when the sound sources are treated as radiating Rayleigh waves in (a) the forward direction only and (b) both forward and backward directions 015005-11 Zhang et al AIP Advances 7, 015005 (2017) FIG Sketch of the angle beam wedge transducers with the wedges covering (a) the entire footprint and (b) a half of the footprint D Rayleigh sound beams with different sizes of wedges It is well known that, by the virtue of reciprocity, acoustic coupling between the incident longitudinal wave in the wedge and the Rayleigh wave on the surface of the solid is the same as that of the surface wave back into the wedge via leakage, therefore the wedge transducer has to be truncated before the end of the acoustic footprint to achieve maximum sensitivity.28 Therefore, the full footprint may be excluded underneath the wedge, as Fig 9(b) shows, and the generated Rayleigh wave fields in this condition is what one cares most about The process of calculating the Rayleigh sound beams can be referred to as the mathematical method in Sec C Rayleigh sound beams on the surface of the specimen generated by two different sizes of the wedge are simulated and compared In this simulation, one of the wedges covers the entire footprint (see Fig 9(a)), while the other one covers only a half of the footprint (see Fig 9(b)) Aside from the wedge sizes, all other parameters used in calculations are the same as those in Sec A To indicate the effect of the wedge sizes on the magnitudes of Rayleigh waves, the values of the z-components in the propagating distance are shown in Fig 10(a) For comparison, the results using the 3-D model are shown in Fig 10(b) Note that these values are normalized by the maximum values in each figure In Fig 10(a), one can see that the amplitudes of Rayleigh waves of the two transducers agree well with each other The reason is that different equivalent lengths Lx are used in the simulations However, we found that in Fig 10(b) the displacement amplitudes of the transducer with a half of the footprint were half of that with the full footprint As a special case, Rayleigh waves can be efficiently generated by a contact transducer located at the edge of the specimen in a certain incident angle without a wedge.29 The proposed model can be employed to explain this behavior and calculate the sound field distributions, while the 3-D model is not suitable for this condition because it only works for the area sound sources The simulation results obtained by the proposed model also agree with the study by Huang, who used both multiple line sources and single line sources to replace the area sound sources and obtained almost the same magnitudes of Rayleigh wave displacements generated by angle beam FIG 10 Comparison of the magnitudes of the z-component displacements generated by different wedge transducers calculated using (a) the proposed model and (b) the 3-D model 015005-12 Zhang et al AIP Advances 7, 015005 (2017) wedge transducers.4 A similar result can also be found in the work by Hurley.9 The reason may be that the Rayleigh beams generated by the sound sources under the wedge in different positions have different phase characteristics, and their magnitudes are affected by these phase differences when the sound beams are superimposed An example to explain this behavior is that the maximum pressure magnitudes of longitudinal waves generated by a large and a small size of circular piston transducers will be same when the initial driving pressures are same From this point of view, the proposed model is more suitable for transducers with different sizes of wedges and has a wider scope of application V SUMMARY AND CONCLUSIONS This research presents a framework for analyzing and modeling the Rayleigh wave sound fields generated by angle beam wedge transducers To make the integral region match with the actual Rayleigh sound sources area under the wedge, the conventional line integral representation has been extended to the surface integral expression using the algorithmic method The developed surface integral method can predict accurately Rayleigh sound beam distributions as sound sources come from contact transducer beams Additionally, stable magnitudes of Rayleigh wave displacements have been obtained with the proposed model Our method is also effective in acquiring the Rayleigh sound fields under the wedge and can be applied in calculating Rayleigh wave sound fields when the wedge sizes varies We emphasize that this model is obtained in an ideal situation in which we did not take into account the leaky energy of Rayleigh waves back into the wedge from underneath it Moreover, we assumed that in the proposed model the entire sound sources under the wedge can 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nonlinearity of materials using Rayleigh waves,” NDT&E International 41, 326–329 (2008) ... analyzing and modeling the Rayleigh wave sound fields generated by angle beam wedge transducers To make the integral region match with the actual Rayleigh sound sources area under the wedge, the... main sound beam that makes them propagate a longer distance.3 Rayleigh waves can be generated by angle beam wedge transducers, comb transducers, interdigital transducers, etc.4–7 Regarding an angle. .. between wave propagation and energy flow, and improvement the accuracy of Rayleigh wave detection, it is necessary to calculate accurate Rayleigh wave sound fields generated by angle beam wedge transducers