= Solution
In the formal stiff-cladding limit with a nonvanishing exterior decay rate, the interface <displacement> tends to zero. The even seam mode then has $\cos(mh)=0$, whereas the odd mode has $\sin(mh)=0$. The positive <wavenumbers> are
$$
\boxed{m_{\mathrm{even},n}=\frac{(n+1/2)\pi}{h},\ n=0,1,\ldots,
\qquad m_{\mathrm{odd},n}=\frac{n\pi}{h},\ n=1,2,\ldots.}
$$
The zero odd index would give an identically zero mode and is excluded. The smallest even transverse <wavenumber> is $\pi/(2h)$, so its lowest branch satisfies
$$
\boxed{\omega^2\geq c_{s1}^2\left(k^2+\frac{\pi^2}{4h^2}\right).}
$$
This is the symmetric-mode cutoff: the actual limiting branches have the discrete listed values of $m$. If the exterior shear speed remains finite, the separate decay requirement $\omega^2<c_{s2}^2k^2$ still applies; the cutoff alone does not replace that guided-wave condition.
Back to article page
