Scattering of anti-plane (SH) waves by a semi-elliptical hill: I—Shallow hill

The paper presents an exact, analytical solution to the boundary-valued problem of the two-dimensional scattering of anti-plane (SH) waves by a shallow, semi-elliptical hill on an elastic half-space, based on the method of wave-function expansion in elliptical coordinate and elliptical cosine half-range expansion. It is expressed in terms of an infinite system of simultaneous linear equations that is later truncated for numerical computation. Some numerical results when semi-elliptical tends to semi-circular are compared with the existing results presented by Lee et al. (2006) [12]. Complicated effects on ground motion due to the existence of an elliptical hill at various aspect ratios and angles of wave incidence are illustrated.

► We model the analytic solution of SH wave scattering by a shallow elliptical hill.
► The Mathieu wave function solution uses the elliptical cosine half-range [0,π] expansion.
► It is expressed as an infinite system of equations to be truncated for computation.
► Numerical results agree with Lee et al. [12] when the elliptical hill is circular.
► Surface motions at shallow hills of various shapes and aspect ratios are presented.