= Solution
Choose depth $z$ positive downwards, the free surface at $z=0$ and the interface at $z=h$. A <Love wave> has only transverse <displacement field> $u_y=U(z)e^{i(\kappa x-\omega t)}$. Take $\omega>0$, $\kappa=\omega/c_L$, and define
$$
q=\sqrt{\omega^2/\beta_0^2-\kappa^2},\qquad p=\sqrt{\kappa^2-\omega^2/\beta^2}.
$$
A trapped mode requires \b[$\beta_0<c_L<\beta$], so both $q,p$ are positive. The SH equations are $U_0''+q^2U_0=0$ in the layer and $U''-p^2U=0$ in the half-space. Zero shear <traction> at the free surface and decay at large depth select
$$
U_0=A\cos(qz),\qquad U=C e^{-p(z-h)}.
$$
Continuity of <displacement field> and shear <traction> at $h$ gives $C=A\cos(qh)$ and $-\mu_0qA\sin(qh)=-\mu pC$. Consequently
$$
\boxed{\tan(qh)=\frac{\mu p}{\mu_0q}.}
$$
Substituting the definitions of $q,p$ yields the requested speed-frequency dispersion relation. For negative <frequency> use $|\omega|$ in both vertical wavenumbers; the real <displacement field> has conjugate <frequency> partners. Neither equality in the trapped-speed interval gives a strictly localized finite-frequency Love mode.
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