= Solution
Substitution of $w_i=f_i(y)e^{i(kx-\omega t)}$ gives
$$
\boxed{f_i''+\left(\frac{\omega^2}{c_{si}^2}-k^2\right)f_i=0.}
$$
The oscillatory seam solution requires $m^2=\omega^2/c_{s1}^2-k^2$, giving
$$
\boxed{\omega^2=c_{s1}^2(k^2+m^2),\qquad
c_L=\frac\omega{|k|}=c_{s1}\sqrt{1+m^2/k^2}.}
$$
For an exponentially decaying exterior, $f_2''=\ell^2f_2$ with $\ell>0$, so the same differential equation instead requires
$$
\boxed{\ell^2=k^2-\omega^2/c_{s2}^2,\qquad
\omega^2=c_{s2}^2(k^2-\ell^2).}
$$
The printed exterior plus sign is inconsistent with its decaying ansatz and must be a sign error. The guided mode has $c_{s1}<c_L<c_{s2}$, so a slower shear-wave seam than its cladding is necessary.
For an even seam mode $f_1=A\cos(my)$, upper-interface <displacement> and <traction> continuity give
$$
A\cos(mh)=A_2e^{-\ell h},\qquad
-\mu_1Am\sin(mh)=-\mu_2\ell A_2e^{-\ell h}.
$$
The lower interface gives the same equations when $B_2=A_2$. Eliminating the amplitudes yields
$$
\boxed{\mu_1m\tan(mh)=\mu_2\ell.}
$$
For an odd seam mode $f_1=B\sin(my)$, the equations are
$$
B\sin(mh)=A_2e^{-\ell h},\qquad
\mu_1Bm\cos(mh)=-\mu_2\ell A_2e^{-\ell h}.
$$
With $B_2=-A_2$ the lower condition agrees, and elimination gives
$$
\boxed{\mu_1m\cot(mh)=-\mu_2\ell.}
$$
These interface conditions together with the two corrected dispersion identities determine the guided-wave branches.
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