Surface shear waves in a half-plane with depth-variant structure

TL;DR

This paper analyzes surface shear wave propagation in a half-plane with depth-dependent shear modulus and density, establishing conditions for the existence and multiplicity of surface wave solutions based on material properties.

Contribution

It introduces criteria for the existence and non-existence of surface shear waves in a depth-variant medium, including the possibility of infinitely many solutions for given wave numbers.

Findings

Criteria for non-existence of surface waves.

Conditions for finite and infinite solutions.

Possibility of infinitely many solutions for any wave number.

Abstract

We consider the propagation of surface shear waves in a half-plane, whose shear modulus $\mu(y)$ and density $\rho(y)$ depend continuously on the depth coordinate $y$. The problem amounts to studying the parametric Sturm-Liouville equation on a half-line with frequency $\omega$ and wave number $k$ as the parameters. The Neumann (traction-free) boundary condition and the requirement of decay at infinity are imposed. The condition of solvability of the boundary value problem determines the dispersion spectrum $\omega(k)$ for the corresponding surface wave. We establish the criteria for non-existence of surface waves and for the existence of $N(k)$ surface wave solutions, with $N(k) \to \infty$ as $k \to \infty$. The most intriguing result is a possibility of the existence of infinite number of solutions, $N(k)=\infty$, for any given $k$. These three options are conditioned by the properties of $\mu(y)$ and $\rho(y)$.