Effective medium description of the resonant elastic-wave response at the periodically-uneven boundary of a half-space

TL;DR

This paper develops an effective medium model for the resonant elastic-wave response at a periodically-uneven boundary of a half-space, revealing dispersive, lossy properties that explain low-frequency resonances.

Contribution

It introduces a novel effective medium approach that simplifies the complex boundary into a dispersive, lossy layer, enabling easier analysis of low-frequency resonances.

Findings

The effective layer exhibits dispersion and loss, unlike the underlying solid.

The model explains Love mode resonance and shear wall pseudo-resonance phenomena.

Resonance characteristics vary predictably with boundary geometry.

Abstract

A periodically-uneven (in one horizontal direction) stress-free boundary covering a linear, isotropic, homogeneous, lossless solid half space is submitted to a vertically-propagating shear-horizontal plane, body wave. The rigorous theory of this elastodynamic scattering problem is given and the means by which it can be numerically solved are outlined. At quasi-static frequencies, the solution is obtained from one linear equation in one unknown. At higher, although still low, frequencies, a suitable approximation of the solution is obtained from a system of two linear equations in two unknowns. This solution is shown to be equivalent to that of the problem of a vertically-propagating shear-horizontal plane body wave traveling in the same solid medium as before, but with a linear, homogeneous, isotropic layer replacing the previous uneven boundary. The thickness of this layer is equal to the vertical distance between the extrema of the boundary uneveness and the effective body wave velocity therein is equal to that of the underlying solid, but the effective shear modulus of the layer, whose expression is given in explicit algebraic form, is different from that of the underlying solid, notably by the fact that it is dispersive and lossy. It is shown that this dispersive, lossy effective layer, overriding the nondispersive, lossless solid half space, gives rise to two distinctive features of low-frequency response: a Love mode resonance and a Fixed-base shear wall pseudo-resonance. This model of effective layer with dispersive, lossy properties, enables simple explanations of how the low-frequency resonance and pseudo-resonance vary with the geometric parameters (and over a wide range of the latter) of the uneven boundary.