A note on three-dimensional scattering and diffraction by a hemispherical canyon–I: Vertically incident plane P-wave

•3D orthogonal spherical wave functions satisfy zero-stress at half-space surface.•Explicit, analytic expressions were derived for the spherical-wave functions.•The most difficult part of the problem before will now be solved analytically.•Waves are calculated at much higher frequencies than all previous cases.•Can be extended to all 3D wave problems at elastic and poro-elastic half spaces.

The three-dimensional scattering by a hemi-spherical canyon in an elastic half-space subjected to seismic plane and spherical waves has long been a challenging boundary-value problem. It has been studied by earthquake engineers and strong-motion seismologists to understand the amplification effects caused by surface topography. The scattered and diffracted waves will, in all cases, consist of both longitudinal (P-) and shear (S-) shear waves. Together, at the half-space surface, these waves are not orthogonal over the infinite plane boundary of the half-space. Thus, to simultaneously satisfy both zero normal and shear stresses on the plane boundary numerical approximation of the geometry and/or wave functions were required, or in some cases, relaxed (disregarded). This paper re-examines this boundary-value problem from the applied mathematics point of view, and aims to redefine the proper form of the orthogonal spherical-wave functions for both the longitudinal and shear waves, so that they can together simultaneously satisfy the zero-stress boundary conditions at the half-space surface. With the zero-stress boundary conditions satisfied at the half-space surface, the most difficult part of the problem will be solved, and the remaining boundary conditions at the finite canyon surface will be easy to satisfy.