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 mu(y) and density ro(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 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)→∞ as k→∞. 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 ro(y).