Research Article  Open Access
Diffraction of Elastic Waves in FluidLayered Solid Interfaces by an Integral Formulation
Abstract
In the present communication, scattering of elastic waves in fluidlayered solid interfaces is studied. The indirect boundary element method is used to deal with this wave propagation phenomenon in 2D fluidlayered solid models. The source is represented by Hankel’s function of second kind and this is always applied in the fluid. Our method is an approximate boundary integral technique which is based upon an integral representation for scattered elastic waves using singlelayer boundary sources. This approach is typically called indirect because the sources’ strengths are calculated as an intermediate step. In addition, this formulation is regarded as a realization of Huygens’ principle. The results are presented in frequency and time domains. Various aspects related to the different wave types that emerge from this kind of problems are emphasized. A near interface pulse generates changes in the pressure field and can be registered by receivers located in the fluid. In order to show the accuracy of our method, we validated the results with those obtained by the discrete wave number applied to a fluidsolid interface joining two halfspaces, one fluid and the other an elastic solid.
1. Introduction
In many areas of physics, the study of fluidsolid interfaces has always attracted interest. For example, important developments to study the dynamic behavior of an oceanic layer over an elastic solid by means of analytical solutions can be seen in the original works of Biot [1] and Ewing et al. [2], where the attention to Stoneley and Rayleigh waves was paid. Other applications have been aimed to understand the behavior of interface waves being focused to ocean bottom [3, 4]. Carcione and Helle [5] studied the physics related to these interfaces in a variety of seabed mechanical properties, from soft sediments to crustal rocks. Analytical results to show the appearance of Rayleigh waves in oceanic ambient excited by deep earthquakes were presented, for instance, in [6, 7].
Attenuation and dispersion of interface waves were investigated in multilayered cases [8–12]. The inverse problem of determining the mechanical parameters of layered media in contact with fluids by measuring the variation of the pressure fields in the fluid was published by Zein et al. [13]. Studies applied to porous media have evidenced the huge influence of porosity in wave propagation, particularly, when the medium is partially saturated [14–18]. Attenuation and dispersion in interface waves due to the presence of fractures were studied in [11, 19–22]. The use of Green’s functions for layered acoustic and elastic formations in 3D was applied in Tadeu and António [23, 24], which can be used in numerical modeling.
In the field of numerical methods such as finite element and finite difference methods, the solution of fluidsolid interfaces was carried out, for instance, in Zienkiewicz and Bettess [25] and Thomas et al. [26], respectively. In addition, spectral element and pseudospectral formulations have revealed to be accurate methods for modeling realistic geometries (see, e.g., Carcione et al. [27]). On the other hand, Komatitsch et al. [28] developed the spectral element method to deal with more complex problems and numerous advantages over classical approaches were remarked.
In this communication, the indirect boundary element method (IBEM) to study the interactions between acoustic and elastic waves, near a fluid and elasticlayered solid interface, is applied. Monopole point source, characterized by Hankel’s function of second kind, is employed to produce an initial pressure wave in the fluid. This formulation could be considered as a numerical realization of Huygens’ principle, in which the diffracted waves are built at the boundary from which they are radiated. Mathematically speaking, this is completely equivalent to the wellknown Somigliana representation theorem. The accuracy of our results for a fluidsolid interface is verified with respect to those obtained by means of the discrete wave number (DWN). Observations previously described by other authors are highlighted. In the following sections, the formulations of both DWN and IBEM applied to fluidsolid interfaces are detailed.
2. Formulation of the Problem by Means of the Discrete Wave Number
The DWN method is one of the techniques to simulate seismic motion. The seismic waves radiated from a source could be expressed as an integral in the wavenumber domain. Moreover, the source is represented as a superposition of homogenous plane waves propagating in discrete angles (see, e.g., Bouchon and Aki [29]). In this method, when denominator of the integrand becomes zero for a particular wavenumber, then the numerical integration could be a difficult task. To overcome this problem, an approach to include complex frequency was suggested as early as the proposal of the DWN method itself. In what follows, a brief description of this method applied to fluidsolid interfaces is shown.
The incident pulse at the fluid, as shown in Figure 1, can be given as whereis incident pulse at the fluid,,is scale coefficient for the incident pulse,is Hankel’s function of second kind and zero order,is circular frequency,is compressional wave velocity in the fluid,is the distance from the receiver to the source (incident pulse),is the wavenumber, andwith Im. If we expressin discrete values, then we haveandwith,wavenumber increment, andis the imaginary unit.
If we consider that the entire pressure and displacement fields in the fluid are expressed as the sum of the free and diffracted fields, then one has whereis density of the fluid,is the distance between the source and the interface, andrepresents the diffraction coefficient for the fluid to be found.
For the solid, we assume that potentials of displacement have the formand, wherewithandwith.andare the compressional and shear wave velocities, respectively. Similarly,andrepresent the diffraction coefficients, for the solid, to be found.
The displacement field for the solid is expressed asand. The stress field can be obtained from the wellknown equation: whereis stress tensor,andare Lamé’s constants,is strain tensor, andis Kronecker’s delta.
The boundary conditions to be applied to the interface are for displacements, and for tractions.
Once the boundary conditions were applied, the unknown coefficients,, andare obtained and then the whole pressure field in the fluid is finally calculated by means of (2). The system of equation to be solved is given by
3. Formulation of the Problem by Means of the Indirect Boundary Element Method
For the IBEM, the incident pulse (source) in the fluid is also represented by (1) and applied as shown in Figure 2(a).
(a)
(b)
Assume that the equation that governs the wave motion in the fluid is given by If we consider that stresses in the fluid can be linked to the pressure generated by the incident pulse, then one can express this as Then, the displacement field can be represented by the wellknown form:
To express the diffracted wave field (for pressure and displacement, resp.) in the fluid due to the source impacting the elastic medium, we propose the use of the following integral representation: where whereis force density for the fluid,is Green’s function for the fluid, andand defines the region orientation (see explanation for, given below).
The complete pressure and displacement field in the fluid, besides free and diffracted ones, can be written, respectively, as Since the source is only applied in the fluid, only diffracted waves appear in the solid and they can be established as follows.
Consider a domainwith a boundary. If the domain is occupied by an elastic solid, the displacement field under harmonic excitation can be expressed, neglecting body forces, by means of the singlelayer boundary integral equation: whereisth component of the displacement at point,is Green’s function, which represent the displacement produced in the directionatdue to the application of a unitary force in directionat point, andis the force density in the directionat point. The productis the force distribution at the surface(the subscriptsare limited to be 1 or 3). The subscript in the differential shows the variable over which the integration is done. This integral equation can be obtained from Somigliana’s representation [30]. Furthermore, it was demonstrated that ifis continuous along, then in that case, the displacement field is continuous across[31].
The integral representation (16) permits the calculation of stresses and tractions by using the direct application of Hooke’s law and Cauchy’s equation, respectively, except for singularities on the boundary, that is, whenis equal toon the surface. From a limiting process established on equilibrium considerations, around an internal vicinity of the boundary, one can write, foron, whereos theth component of tractions,iftends to the boundary“from inside” the region,iftends to“from outside” the region, orifis not at.is the traction Green’s function, that is, the traction in the directionat a point, linked to the unit vector, due to the application of a unitary force in the directionaton. The 2D Green’s functions for infinite spaces can be seen in [32, 33].
3.1. Boundary Conditions
From the configuration depicted in Figure 2(b), it is convenient to partition the domain in three regions (,, and), for which proper boundary conditions should be established. These conditions for fluidsolid interfaces can be written as follows.
For the fluidsolid interface,
For the continuous solid interface, Expressing (18) as a function of the diffracted field (16) for the solid, and incident and diffracted fields ((10) and (12), resp.) for the fluid, one obtains: The traction free condition (19) is expressed from its integral form (17), resulting in The Equation (20) can be expressed by means of (17), (1), and (11), and then one has Equations (21) express the continuity condition that must exist between the interface of regionsand(boundariesand). These are defined as follows:
3.2. Discretization
Here, the discretization of (22) to (26) is presented. Considering that force densitiesandare constant on each boundary element that forms the surfaces of regions,, and(see Figure 2(b)) and Gaussian integration (or analytical integration, where Green’s function is singular) is performed, (22) is rewritten as,
Equation (23) leads to
Equation (24) can be written as
Equations (25) and (26) lead to where Equations (27), (30), (32), and (34) form a system of integral equations to be solved. Once the force densities (and) were obtained, the whole displacement and pressure fields in the fluid are found by means of (14) and (15), respectively. For the solid, the complete displacement and traction fields are calculated applying (16) and (17), respectively. In the following section, we verify the accuracy of the IBEM and DWN for several cases applied to an interface that joins two halfspaces, one fluid and the other solid. Moreover, we present results for layered models using IBEM. Additional details on the discretization scheme can be consulted in [35–38]. Work related to fluidsolid interfaces using IBEM was also presented in [39], in which general aspects of wave propagation in fluidsolid media were pointed out. The novelty of the present paper is related to the application of the IBEM to model wave propagation in layered solidfluid systems. In this sense, this paper deals specifically with fluidlayered solids and provides the following aspects.(a)It contains the complete formulation applied to fluidlayered solid interfaces. Moreover, the paper describes the precise boundary conditions and discretization scheme for fluidlayered solid media.(b)It contains pressure spectra that clearly show the influence of a fluidlayered medium, according to its layer thickness.(c)It presents synthetic seismograms of pressures for several interfaces like fluidsandstonegranite interfaces and others.(d)It pointed out the kind of interface waves that emerge in each case.(e)Finally, the discussions focused on the diffracted waves that appear due to the presence of the layer.
These results are shown in the following sections.
4. Validation and Application of the IBEM and DWN Methods
To verify the accuracy of both formulations (IBEM and DWN), we considered various models with fluidsolid interfaces. Borejko [34] developed theoretical and experimental studies in order to show the emergence of interface waves. The material properties for his models are described in Table 1.

For comparison purposes, we chose an interface model joining two halfspaces, one fluid and the other an elastic solid, for casesandshown in Table 1.
As shown in Figure 1, one receiver located at the distances m and m was considered. In Figure 3, the spectrum of pressure for such receiver is shown for casesand. Results from IBEM are plotted with solid and dashed lines to represent Limestone and Pitch, respectively. Calculations from DWN are depicted with circles for case and asterisks for case. Good agreement can be appreciated between IBEM and DWN results. It is clear from this figure that the responses for both materials vary significantly and can be associated with the relative value of the shear wave velocity of the solid in comparison with the fluid wave velocity [34]. For shear wave velocities higher than the compressional wave velocity for the fluid, the spectrum of pressures shows more simple patterns in comparison with the opposite case, where some resonant peaks are observed.
In Figure 4, spectra of pressures for four cases (to) solved by IBEM are displayed. For the casesand, the thickness of the layer is m and m, for both cases. For these layered models, h refers to the layer thickness (e.g., sandstone is the layer for case, while granite is the layer for case). Casesandcorrespond to simple models (let us refer to them as simple models), in other words, an interface that joins two halfspaces, one fluid and the other solid. These cases are plotted using dotted lines (case) and circles (case), both shown in Figures 4(a) and 4(b). In Figure 4(a), the response for caseis also included (for m and m).
(a)
(b)
An interesting fact that emerges from the results observed in Figure 4(a) is that the behavior shown by the layered model (sandstonegranite) is delimited by the simple models (granite and sandstone). Besides, at low frequencies the layered model behavior shows a clear tendency to case. In other words, the material that constitutes the halfspace (see regionin Figure 2(b)) controls this phenomenon (i.e., granite for caseand sandstone for case). At high frequencies the behavior is controlled by the shallow material (see regionin Figure 2(b)) due to shorter wavelengths, and then the material that constitutes the halfspace has no influence in the response.
Analogously, the behavior shown by caseis delimited by the simple models (casesand 4). At low frequencies, its response is very close to that obtained by case. In contrast, at high frequencies, such response is near to case(granite). For both figures (Figures 4(a) and 4(b)), there is a transition zone, which depends on the layer thickness ().
In Figure 5, pressure fields in time domain are shown for cases,, and(see Table 1). To this end, a fast Fourier transform (FFT) algorithm was applied using a Ricker wavelet as source time function (i.e., the second derivative of a Gaussian) with a characteristic period of sec; this source is used in all analyses presented here. All models were analyzed for frequency increments of 150 Hz, reaching a final frequency of 19200 Hz. By means of the Fourier transformation, it is possible to observe different kinds of waves that emerge.
(a)
(b)
(c)
(d)
For all cases, 25 receivers were located in the fluid. The first one was placed in a distance of m from the source, and the rest of the receivers were placed using a distance increment of 0.05 m. The distance m (see Figure 2) for all cases. For case(fluidsandstonegranite interface), two layer thicknesses were considered, which are m (Figure 5(c)) and m (Figure 5(d)).
In Figure 5(a), the wave propagation phenomenon for case(fluidsandstone interface) is shown. Here, it is possible to observe the influence of the(compressional wave velocity of sandstone, also known aswave velocity) represented as; the direct wave that travels in the fluid and that is perceived by the receivers is shown withand the Scholte’s interface wave is illustrated using. The super indexrepresents “Sandstone,” whileis for “Fluid.” Borejko [34] also found this kind of waves by means of theoretical and experimental studies. For this case, the velocities measured werems^{−1},ms^{−1}, and ms^{−1}.
In Figure 5(b), the wave fronts that emerge from the fluidgranite interface interactions are depicted. Here, it is possible to identify wave velocities associated with pseudo Rayleigh, direct and Scholte’s waves, which propagate at ms^{−1}, ms^{−1}, and ms^{−1}, respectively. The super indexrefers to “Granite.” Our results for these last two cases agree with those obtained by Borejko [34]. It is important to mention that Scholte’s wave travels at a velocity close to the direct wave in the fluid and, then, only one wave front is seen. This was also reported by Borejko [34].
For the interface model (case), two layer thicknesses were studied, as mentioned above. In this case, the influence of granite is evident. In Figure 5(c) (for m), the direct wave represented byand the four fronts associated with Scholte’s wave velocity () are clearly observed. Moreover, two wave fronts that travel at a velocity of ms^{−1} are identified. This velocity coincides with(shear wave velocity for sandstone, also known aswave velocity). The subindexstands forwave velocity. In Figure 5(d), these same wave fronts appear, but less interactions are evidently appreciated due to the considered layer thickness ( m).
5. Conclusions
The indirect boundary element method to study the wave propagation phenomenon in 2D fluidlayered solid interfaces was used. This indirect formulation can give to the analyst a deep physical insight on the generated diffracted waves because it is closer to the physical reality and can be regarded as a realization of Huygens’ principle. In any event, it is fully equivalent to the classical Somigliana representation theorem. In order to verify the accuracy, we tested our method by comparing results with the analytical solution known as discrete wave number. A near interface pulse generates scattered waves that can be registered by receivers located in the fluid. Results were presented in frequency and time domains, where some aspects related to different wave types that emerge from this kind of problems were pointed out. The results between IBEM and DWN describe the same physics and are, for engineering purposes, quite approximated.
Acknowledgments
Thanks are given to G. Sánchez, E. Plata, E. Chávez, and L. A. Cruz of Unidad de Servicios de Información (USI) of Instituto de Ingeniería, UNAM; their help was crucial to locate useful references. Partial supports from DGAPAUNAM, Project IN121709, SENERCONACYT Project 128376, and from the Instituto Mexicano del Petróleo are greatly appreciated. This work has been partial supported by the National Polytechnic Institute of México under project SIP20130622.
References
 M. A. Biot, “The interaction of Rayleigh and Stoneley waves in the ocean bottom,” Bulletin of the Seismological Society of America, vol. 42, pp. 81–93, 1952. View at: Google Scholar  MathSciNet
 W. M. Ewing, W. S. Jardetzky, and F. Press, Elastic Waves in Layered Earth, McGrawHill, New York, NY, USA, 1957. View at: MathSciNet
 W. L. Roever, J. H. Rosenbaum, and T. F. Vining, “Acoustic waves from an impulsive source in a fluidfilled borehole,” Journal of the Acoustical Society of America, vol. 55, no. 6, pp. 1144–1157, 1974. View at: Publisher Site  Google Scholar
 R. van Vossen, J. O. A. Robertsson, and C. H. Chapman, “Finitedifference modeling of wave propagation in a fluidsolid configuration,” Geophysics, vol. 67, no. 2, pp. 618–624, 2002. View at: Publisher Site  Google Scholar
 J. M. Carcione and H. B. Helle, “The physics and simulation of wave propagation at the ocean bottom,” Geophysics, vol. 69, no. 3, pp. 825–839, 2004. View at: Publisher Site  Google Scholar
 M. Yoshida, “Velocity and response of higher mode Rayleigh waves for the Pacific Ocean,” Bulletin of the Earthquake Research Institute, vol. 53, pp. 1135–1150, 1978. View at: Google Scholar
 M. Yoshida, “Group velocity distributions of Rayleigh waves and two upper mantle models in the Pacific Ocean,” Bulletin of the Earthquake Research Institute, vol. 53, pp. 319–338, 1978. View at: Google Scholar
 I. Karasalo, “Exact finite elements for wave propagation in rangeindependent fluidsolid media,” Journal of Sound and Vibration, vol. 172, no. 5, pp. 671–688, 1994. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 M. Schoenberg, “Wave propagation in alternating solid and fluid layers,” Wave Motion, vol. 6, no. 3, pp. 303–320, 1984. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 A. H. Nayfeh, T. W. Taylor, and D. E. Chimenti, “Theoretical wave propagation in multilayered orthotropic media,” in Proceedings of the Joint Wave Propagation in Structural Composites on Wave Propagation in Structural Composites, vol. 90 of AMD, pp. 17–27, June 1988. View at: Google Scholar
 T. Xia, H. Chen, and S. Wu, “Further discussion on Rayleigh wave characteristics in layered fluidsolid media,” Journal of Vibration Engineering, vol. 12, pp. 348–353, 1999. View at: Google Scholar
 B. Gurevich and R. Ciz, “Shear wave dispersion and attenuation in periodic systems of alternating solid and viscous fluid layers,” International Journal of Solids and Structures, vol. 43, no. 2526, pp. 7673–7683, 2006. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 S. Zein, E. Canot, J. Erhel, and N. Nassif, “Determination of the mechanical properties of a solid elastic medium from a seismic wave propagation using two statistical estimators,” Mathematics and Mechanics of Solids, vol. 13, no. 5, pp. 388–407, 2008. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 M. J. Mayes, P. B. Nagy, L. Adler, B. P. Bonner, and R. Streit, “Excitation of surface waves of different modes at fluidporous solid interfac,” Journal of the Acoustical Society of America, vol. 79, no. 2, pp. 249–252, 1986. View at: Publisher Site  Google Scholar
 J. Wang, C. Zhang, and F. Jin, “Analytical solutions for dynamic pressures of coupling fluidsolidporous medium due to P wave incidence,” Earthquake Engineering and Engineering Vibration, vol. 3, no. 2, pp. 263–271, 2004. View at: Publisher Site  Google Scholar
 Y. Liu and K.X. Liu, “Characteristic analysis for stress wave propagation in transversely isotropic fluidsaturated porous media,” Applied Mathematics and Mechanics (English Edition), vol. 25, no. 6, pp. 656–663, 2004. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 J. N. Sharma and V. Pathania, “Propagation of leaky surface waves in thermoelastic solids due to inviscid fluid loadings,” Journal of Thermal Stresses, vol. 28, no. 5, pp. 485–519, 2005. View at: Publisher Site  Google Scholar
 I. Malinouskaya, V. V. Mourzenko, J.F. Thovert, and P. M. Adler, “Wave propagation through saturated porous media,” Physical Review E, vol. 77, no. 6, Article ID 066302, 2008. View at: Publisher Site  Google Scholar  MathSciNet
 V. V. Varadan, T. A. K. Pillai, and V. K. Varadan, “Ultrasonic wave scattering by a subsurface flaw in joined fluidsolid half spaces,” Journal of Applied Mechanics, Transactions ASME, vol. 50, no. 4, pp. 802–806, 1983. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 X. M. Tang, C. H. Cheng, and M. N. Toksoz, “Stoneleywave propagation in a fluidfilled borehole with a vertical fracture,” Geophysics, vol. 56, no. 4, pp. 447–460, 1991. View at: Publisher Site  Google Scholar
 A. S. Eriksson, A. Boström, and S. K. Datta, “Ultrasonic wave propagation through a cracked solid,” Wave Motion, vol. 22, no. 3, pp. 297–310, 1995. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 A. S. Ashour, “Propagation of guided waves in a fluid layer bounded by two viscoelastic transversely isotropic solids,” Journal of Applied Geophysics, vol. 44, no. 4, pp. 327–336, 2000. View at: Publisher Site  Google Scholar
 A. Tadeu and J. António, “2.5D Green's functions for elastodynamic problems in layered acoustic and elastic formations,” Journal of Computer Modeling in Engineering and Sciences, vol. 2, no. 4, pp. 477–495, 2001. View at: Google Scholar  Zentralblatt MATH
 A. Tadeu and J. M. P. António, “Acoustic insulation of single panel walls provided by analytical expressions versus the mass law,” Journal of Sound and Vibration, vol. 257, no. 3, pp. 457–475, 2002. View at: Publisher Site  Google Scholar
 O. C. Zienkiewicz and P. Bettess, “Fluidstructure dynamic interaction and wave forces, an introduction to numerical treatment,” International Journal for Numerical Methods in Engineering, vol. 13, no. 1, pp. 1–16, 1978. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 C. Thomas, H. Igel, M. Weber, and F. Scherbaum, “Acoustic simulation of Pwave propagation in a heterogeneous spherical earth: Numerical method and application to precursor waves to PKPdf,” Geophysical Journal International, vol. 141, no. 2, pp. 307–320, 2000. View at: Publisher Site  Google Scholar
 J. M. Carcione, H. B. Helle, G. Seriani, and M. P. Plasencia Linares, “Simulation of seismograms in a 2D viscoelastic Earth by pseudospectral methods,” Geofisica Internacional, vol. 44, no. 2, pp. 123–142, 2005. View at: Google Scholar
 D. Komatitsch, C. Barnes, and J. Tromp, “Wave propagation near a fluidsolid interface: a spectralelement approach,” Geophysics, vol. 65, no. 2, pp. 623–631, 2000. View at: Publisher Site  Google Scholar
 M. Bouchon and K. Aki, “Discrete wave number representation of seismic source wave fields,” Bulletin of the Seismological Society of America, vol. 67, pp. 259–277, 1977. View at: Google Scholar
 F. J. SanchezSesma and M. Campillo, “Diffraction of P, SV and Rayleigh waves by topographic features: a boundary integral formulation,” Bulletin of the Seismological Society of America, vol. 81, no. 6, pp. 2234–2253, 1991. View at: Google Scholar
 V. D. Kupradze, “Dynamical problems in elasticity,” in Progress in Solid Mechanics, I. N. Sneddon and R. Hill, Eds., vol. 3, NorthHolland, Amsterdam, The Netherlands, 1963. View at: Google Scholar
 A. RodríguezCastellanos, F. Luzón, and F. J. SánchezSesma, “Diffraction of seismic waves in an elastic cracked halfplane using a Boundary Integral Formulation,” Soil Dynamics and Earthquake Engineering, vol. 25, pp. 827–837, 2005. View at: Publisher Site  Google Scholar
 A. RodríguezCastellanos, F. J. SánchezSesma, F. Luzón, and R. Martin, “Multiple scattering of elastic waves by subsurface fractures and cavities,” Bulletin of the Seismological Society of America, vol. 96, pp. 1359–1374, 2006. View at: Publisher Site  Google Scholar
 P. Borejko, “A new benchmark solution for the problem of watercovered geophysical bottom,” in Proceedings of the International Symposium on Mechanical Waves in Solids, Zhejiang University, Hangzhou, China, 2006. View at: Publisher Site  Google Scholar
 A. RodríguezCasteiianos and F. J. SánchezSesma, “Numerical simulation of multiple scattering by hidden cracks under the incidence of elastic waves,” Advanced Materials Research, vol. 65, pp. 1–8, 2009. View at: Publisher Site  Google Scholar
 A. RodríguezCastellanos, R. AvilaCarrera, and F. J. SánchezSesma, “Scattering of elastic waves by shallow elliptical cracks,” Revista Mexicana de Física, vol. 53, pp. 254–259, 2007. View at: Google Scholar
 A. RodríguezCastellanos, R. ÁvilaCarrera, and F. J. SánchezSesma, “Scattering of Rayleighwaves by surfacebreaking cracks: an integral formulation,” Geofísica Internacional, vol. 46, pp. 241–248, 2007. View at: Google Scholar
 R. AvilaCarrera, A. RodríguezCastellanos, F. J. SánchezSesma, and C. OrtizAlemán, “Rayleighwave scattering by shallow cracks using the indirect boundary element method,” Journal of Geophysics and Engineering, vol. 6, pp. 221–230, 2009. View at: Publisher Site  Google Scholar
 A. RodríguezCastellanos, E. Flores, F. J. SánchezSesma, C. OrtizAlemán, M. NavaFlores, and R. Martin, “Indirect boundary element method applied to fluidsolid interfaces,” Soil Dynamics and Earthquake Engineering, vol. 31, no. 3, pp. 470–477, 2011. View at: Publisher Site  Google Scholar
Copyright
Copyright © 2013 J. E. BasaldúaSánchez et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.