Scattering and diffraction ofSHwaves by a finite crack: an analytical solution

Abstract

Summary The diffraction of SH waves by a finite plane crack is studied. The classical Sommerfeld solution for a semi‐infinite straight reflecting screen is used as a building block to calculate the diffracted field generated by a finite crack. The solution is derived from the analysis of the behaviour of diffracted waves. These waves, which are first generated at the edges of the crack, travel along the surfaces and are diffracted/reflected at the opposite edge. By iteratively taking into account the contribution to the total field of these travelling waves, an infinite series with a known limit is constructed, leading to an approximate analytical solution for the case of a finite plane crack. This solution is virtually exact for large frequencies and it is very good for incoming wavelengths of up to four times the size of the crack. Since the solution is explicit the computational cost is very low. Both frequency and time‐domain results are included.

The scattering of SH waves by a finite crack with a superposition-based diffraction technique

“…On the other hand, the validation in the time domain was conducted in terms of synthetic seismograms, also for receivers over the shadow zone and in terms of snapshots of the propagation patterns over the complete computational domain. As an additional validation we also compared our results with those obtained by Sánchez-Sesma & Iturrarán-Viveros (2001), where the diffracted waves were assumed as plane fronts neglecting its cylindrical nature. The analyses were first conducted in the frequency domain, where we computed the transfer functions between the total response and the incident wave.…”

“…That work was also relevant since it opened the door to a number of studies dealing with more general cases. For instance Sánchez-Sesma & Iturrarán-Viveros (2001), performing superposition of Sommerfeld's solution, found an analytic expression for the diffracted field produced by of a plane SH wave incident upon a single crack of finite size. In that solution the first diffracted waves generated at the tips were limited to propagate as plane waves when interacting with the opposite tip of the crack, thus neglecting its original cylindrical nature.…”

“…In other words, in the superposition based diffraction technique there is a direct relation among the infinite terms in the series and actual diffraction events and the number of required terms can be shown to be frequency dependent. In this work we use this superposition approach and construct the solution like in Sánchez-Sesma & Iturrarán-Viveros (2001), where the finite crack is obtained as the superposition of two semi-infinite cracks. While these authors used as starting point the fundamental solution by Sommerfeld (1896), accounting only by the diffraction of plane waves, we started from the generalized wedge solved by Kouyoumjian & Pathak (1974) which directly considers the initially diffracted waves as cylindrical fronts.…”

Sensitivity analysis of fracture scattering

“…In field reservoirs, fractures always have finite length, and fractures with characteristic lengths on the order of seismic wavelength are one of the important scattering sources that generate seismic codas. Sanchez-Sesma and Iturraran-Viveros (2001) derived an approximate analytical solution of scattering and diffraction of SH waves by a finite fracture, and Chen (submitted to SEG 2010) derived an analytical solution for scattering from a 2D elliptical crack in an isotropic acoustic medium. However, so far it is still too difficult to derive the…”

The simulation of 3D elastic scattering produced by thin rigid inclusions using the traction boundary element method

“…Some of the first analytical studies on wave diffraction and scattering focused on the wave motion and reverberations in alluvial basins of regular shape [9,10]. Other work has examined the wave scattering induced by cylindrical circular inclusions and single plane cracks [11][12][13][14][15][16]. Since these analytical approaches are only known for simple and regular geometries, most researchers have concentrated on developing numerical schemes.…”