 Guided shear modes of a symmetric embedded elastic layer

For an isotropic elastic layer $|y|<h$ between two identical elastic half-spaces, an antiplane displacement $w=f(y)e^{i(kx-\omega t)}$ has $m^2={\omega }^2/c_{s1}^2-k^2>0$ in the layer and exterior decay rate ${\ell }^2=k^2-{\omega }^2/c_{s2}^2>0$. Hence a trapped mode has phase speed between the layer and cladding shear speeds. Continuity of displacement and shear traction gives the first displayed equation for even modes $f\propto \cos (my)$ and the second for odd modes $f\propto \sin (my)$.

In a rigid-cladding limit with nonzero decay rate, the interfaces become displacement nodes. Even transverse wavenumbers are $(n+1/2)\pi /h$, $n\ge 0$, and odd ones are $n\pi /h$, $n\ge 1$. The minimum even-mode dispersion branch is ${\omega }^2=c_{s1}^2[k^2+{\pi }^2/(4h^2)]$. When the cladding speed is finite, exterior decay remains a separate requirement; satisfying this lower cutoff alone is insufficient.