S U M M A R YA 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.

S U M M A R YThe scattering problem of plane SH waves by convex circular-arc topography is investigated. The series solution corresponding to this problem is derived via the wavefunction expansion method, combining with the region-matching technique. For the semi-circular case, the proposed formulation can be reduced to a simpler form than that given in the literature and consequently is easier to calculate. Both the steady-state and transient responses for some parameters are evaluated and discussed. Besides, based on the theory of geometric optics, for a concave mirror the focusing of light will take place, implying that analogous phenomenon may arise in the case of present topography. Such a focusing phenomenon will cause localized concentration of high amplitude of wave motion, which may have a great influence on underground structures. Therefore, the focusing effect by convex circular-arc surface is further examined in both the frequency and time domains. Numerical results show that the geometry under consideration is indeed capable of focusing wave energy, which is a significant contributor to the amplification behaviour .

S U M M A R YBased on the application of the region-matching technique, an analytical approach is presented for the scattering of plane SH waves from a shallow symmetrical V-shaped canyon, and then a series solution is derived. The analysed region is divided into an enclosed and an open region by introducing a semi-circular auxiliary boundary. In each region, the displacement field can be expressed as infinite sum of appropriate wavefunctions satisfying partial boundary conditions, respectively. The unknown coefficients can be determined by enforcing the continuity conditions in connection with the Graf's addition formula. The frequency-and time-domain responses are both evaluated and displayed for several physical parameters. From graphical results, the effects of the canyon depth on surface ground motion are conspicuous. The proposed series solutions can serve as benchmark for numerical methods, in particular for those at much higher frequencies.

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S U M M A R YA series solution of the plane SH-waves incident on a partially filled semi-circular alluvial valley imbedded in a half-space is presented. Based on the region-matching method, the analysed region is decomposed into two subregions by the interface between two media. The antiplane displacement field of each subregion is expressed in terms of an infinite series of cylindrical wavefunctions with unknown expansion coefficients. After imposing the traction-free condition on the curved valley surface and the matching conditions on the interface with the aid of Graf's addition theorem, the unknown coefficients are obtained. Both the frequency-and time-domain responses are evaluated. In the theoretical derivation of this work, two classical exact series solutions are also included, so the present series solution is more general than those given before. Visible effects of different physical parameters on ground surface motions are illustrated in graphical form.