  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 $\rho(y)$.
