Surface Shear Waves in a Functionally Graded Half-Space

We analyze the existence and number of solutions of the boundary-value problem describing time-harmonic surface shear waves on the free boundary of a functionally graded half-space with density and shear modulus depending continuously on the semi-infinite depth coordinate. The mathematical formulation amounts to the parametric Sturm-Liouville equation on a half-line with frequency and wave number as the parameters, which is subjected to the Neumann (traction-free) condition at the origin and the condition of vanishing at infinity. The solvability requirement determines the wave number versus frequency dispersion dependence of the surface waves. We establish the criteria for non-existence of surface waves and for the existence of a finite number of surface wave solutions, which grows and tends to infinity with growing wave number. An interesting result is a possibility of the existence of infinite number of solutions for any given wave number. These three options are conditioned by the asymptotic behavior of the shear modulus and density close to the infinite depth. The paper ends up with an overview of possible extensions of the present study in the physical and mathematical context.