Point force excitation of surface waves along the doubly corrugated traction-free boundary of an elastic half-space

Abstract

The excitation of generalized Rayleigh waves due to the effect of a buried time-harmonic source located below the infinite, doubly periodically corrugated free boundary of a homogenous, isotropic, and linear elastic half-space is considered. The full elastodynamic equations are solved using the null-field approach (also called the T matrix method), and the contribution of the surface waves to the total displacement field in the vicinity of the boundary is obtained for the sinusoidally corrugated geometry with equal periods of corrugation. For different corrugation heights and frequencies and both a vertically and a horizontally directed point force excitation, numerical results are presented for the angular dependence of the surface field far away from the source where the generalized Rayleigh mode contribution dominates. When dealing with the horizontally directed point force the direction of the excitation in the horizontal plane is an important factor and it is explored numerically as well. In general, the far-field amplitude increases both with increasing frequency and corrugation height. It is smaller for a horizontal point force than a vertical one with the same location. As expected, the angular dependence of the amplitude is closely connected to the corresponding resonance and damped structure of the surface modes. Also, the appearance of stopbands or damped parts in the surface mode is followed by a disappearance of the far-field amplitude. The numerical computations are affected by some numerical instabilities that seem difficult to circumvent. Instabilities seem to arise when a residue of an inverse matrix is computed numerically and appear for all frequencies and corrugation heights for surface waves propagating at 0°, 45°, and 90° to the corrugations. Others appear only for certain frequencies and corrugation heights at other directions of surface wave propagation and seem to be a consequence of a cross-over resonance between the surface modes.