Surface shear waves in a half-plane with depth-variant structure
We consider the propagation of surface shear waves in a half-plane, whose shear modulus $μ(y)$ and density $ρ(y)$ depend continuously on the depth coordinate $y$. The problem amounts to studying the parametric Sturm-Liouville equation on a half-line with frequency $ω$ 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 $ω(k)$ 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 $μ(y)$ and $ρ(y)$.