An analytical approach for the scattering of SH waves by a symmetrical V-shaped canyon: deep case

A novel analytical approach to the SH-waves scattering problem of a single deep symmetrical V-shaped canyon is presented. The adopted strategy of domain decomposition prevents the auxiliary boundary from being pierced by the lowest part of the canyon, and inherently encompasses the singular behaviour of the stress field around the bottom of the canyon. Appropriate wavefunctions and Graf's addition formulas are well utilized. The introduction of the method of images fulfills the stress-free condition at the ground level. In the deep V-shaped cases, comparisons with previously published data show good agreement. In the degenerate cases where the width of the canyon approaches zero, the presented results coincide with those obtained from the exact series solution of a single zero-thickness vertical edge crack. Both frequency- and time-domain results are given. Effects of the parameters on steady-state surface motions are illustrated and discussed. Transient changes in surface and subsurface displacement fields are included. The proposed series solution not only provides reliable results sufficiently under high-frequency excitations, but also fills the gap in the preceding cases of shallow canyons.