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.