= Guided shear modes of a symmetric embedded elastic layer
{title2=$\mu_1m\tan(mh)=\mu_2\ell,\quad\mu_1m\cot(mh)=-\mu_2\ell$}
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 $(n+1/2)\pi/h$, $n\geq0$, and odd ones are $n\pi/h$, $n\geq1$. 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.
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