= Solution
The full <Love wave> problem replaces the rigid base by a decaying substrate displacement. With $Y=A\cos(qz)$ in the layer and $Y=B e^{-\kappa(z-h)}$ below it,
$$
q^2=\frac{\omega^2}{\beta'^2}-k^2,\qquad
\kappa^2=k^2-\frac{\omega^2}{\beta^2}>0,
\qquad\tan(qh)=\frac{\mu\kappa}{\mu'q},
$$
where $\mu'=\rho'\beta'^2$ and $\mu=\rho\beta^2$. Displacement continuity gives $B=A\cos(qh)$ and <traction> continuity gives $\mu'q\sin(qh)=\mu\kappa\cos(qh)$. The rigid base is therefore a good approximation only when $\mu\kappa\gg\mu'q$: then the substrate displacement is small and $qh$ lies close to $\chi_n=(n+1/2)\pi$.
Put $\epsilon=\beta'/\beta$, $\eta=\mu'/\mu$, $y=qh$, $W=\omega h/\beta'$ and $K=kh$. The complete dispersion branch can be parameterized without root finding:
$$
\boxed{W(y)=\frac{y\sqrt{1+\eta^2\tan^2y}}{\sqrt{1-\epsilon^2}},\qquad
K(y)=\frac{y\sqrt{\epsilon^2+\eta^2\tan^2y}}{\sqrt{1-\epsilon^2}},
\quad n\pi<y<\chi_n}.
$$
Then $c=\beta'W/K$ and $U=\beta'W'/K'$. At the lower end, $c\to\beta$ and $U\to\beta$. The <Love-wave cutoff frequencies> are $\omega_{c,n}=n\pi\beta'/[h\sqrt{1-\epsilon^2}]$, with zero cutoff for the fundamental. Thus the rigid-layer formula cannot describe the fundamental's low-<frequency> limit or the low-<frequency> part of an overtone: it would either give no propagating mode or predict $c>\beta$, outside the trapping interval $\beta'<c<\beta$.
For comparable densities and a fixed mode index, $\eta=(\rho'/\rho)\epsilon^2\ll1$. Write $y=\chi_n-\delta$ near the strong-dispersion transition. In the region controlling the <group-velocity minimum of a high-contrast Love wave>, $\delta\ll1$ and $\eta/\delta\gg\epsilon$. The leading derivatives give
$$
\frac U{\beta'}\simeq\frac{\delta^2}{\eta\chi_n}+\frac\eta\delta.
$$
Their minimum is at
$$
\delta_{\min}\sim\left(\frac{\eta^2\chi_n}{2}\right)^{1/3},\qquad
\boxed{\omega_{\min,n}\sim\frac{\beta'\chi_n}{h}=\omega_n},
\qquad
\frac{U_{\min,n}}{\beta'}\sim\frac3{2^{2/3}}\left(\frac\eta{\chi_n}\right)^{1/3}.
$$
The <frequency> estimate follows by inserting $\delta_{\min}$ into $W\simeq(\chi_n-\delta)[1+\eta^2/(2\delta^2)+\epsilon^2/2]$: the two leading terms of size $\delta$ cancel. The real <group velocity> has a finite minimum and turns upwards near the rigid-layer threshold, whereas the rigid formula has zero group speed at its artificial threshold.
For a useful estimate of the validity range, put $\Delta=W-\chi_n>0$. The rigid prediction has $K_R^2=W^2-\chi_n^2\simeq2\chi_n\Delta$. The finite-load phase correction is $\delta\simeq\eta\chi_n/(\kappa h)$; its changes to both $K$ and $dK/dW$ are small if
$$
\boxed{\Delta\gg\max\left\{\chi_n\epsilon^2,(\eta^2\chi_n)^{1/3}\right\}}.
$$
The first condition keeps $K_R$ away from the substrate propagation boundary; the second controls the rapidly changing interface phase. Equivalently, the rigid formulas are reliable sufficiently above the actual group-velocity minimum, beyond its narrow transition interval. At fixed $\omega/\omega_n>1$ separated from one, they become asymptotically accurate as the contrast grows. They are particularly simple at $\omega\gg\omega_n$, where both velocities approach $\beta'$. Within the transition and below it the full <dispersion relation> is required.
The small speed ratio alone is insufficient if the layer density grows so strongly that $\mu'/\mu$ is not small. For such materials the exact load criterion $\mu\kappa\gg\mu'q$ and its <frequency> derivatives should be used; the preceding contrast estimates assume a bounded order-one density ratio and fixed $n$.
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