 Rigid-backed shear-wave dispersion

A layer with shear modulus ${\mu }_1$ and wave speed $c_1$ lies between a rigid support and a half-space with shear modulus ${\mu }_2$ and wave speed $c_2>c_1$. A trapped antiplane shear normal mode has layer amplitude $A\sin (my)$ and exterior amplitude $A\sin (mh)e^{-\ell (y-h)}$, where $m=k\sqrt{c^2/c_1^2-1}$ and $\ell =k\sqrt{1-c^2/c_2^2}$. Continuity of displacement field and traction gives the displayed dispersion relation. It requires $c_1<c<c_2$ and differs from a Love wave with a free boundary.