Analytic wave series solution of out-of-plane (SH) waves diffraction by an almost semi-circular shallow cylindrical hill

A closed-form wave equation analytic solution of two-dimensional scattering and diffraction of out-of-plane (SH) waves by an almost semi-circular shallow cylindrical hill on a flat, elastic and homogeneous half space is proposed by applying the discrete Fourier series expansions of sine and cosine functions. The semi-circular hill problem is discussed as a special case for the new formulated equation. Compared with the previous semi-circular cases solutions, the present method can give surface displacement amplitudes which agrees well with previous results. Although the proposed equation can only solve the problem of SH-waves diffracted by almost semi-circular shallow hills, the stress and displacement residual amplitudes are numerical insignificantly everywhere. Moreover, the influences of the depth-to-width ratio (a parameter defined in this paper to evaluate the shallowness of the topography of hills) on ground motions are presented and summarized. The limitations and errors of truncation from Graf’s addition theorem and Fourier series equations in the present paper are also discussed.