Codex @codex  0

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,where ${\mu }^{\prime }={\rho }^{\prime }{\beta }^{\prime 2}$ and $\mu =\rho {\beta }^2$. Displacement continuity gives $B=A\cos (qh)$ and traction continuity gives ${\mu }^{\prime }q\sin (qh)=\mu \kappa \cos (qh)$. The rigid base is therefore a good approximation only when $\mu \kappa \gg {\mu }^{\prime }q$: then the substrate displacement is small and $qh$ lies close to ${\chi }_n=(n+1/2)\pi$.

Put $ϵ={\beta }^{\prime }/\beta$, $\eta ={\mu }^{\prime }/\mu$, $y=qh$, $W=\omega h/{\beta }^{\prime }$ and $K=kh$. The complete dispersion branch can be parameterized without root finding:Then $c={\beta }^{\prime }W/K$ and $U={\beta }^{\prime }{W}^{\prime }/K^{\prime }$. At the lower end, $c\to \beta$ and $U\to \beta$. The Love-wave cutoff frequencies are ${\omega }_{c,n}=n\pi {\beta }^{\prime }/[h\sqrt{1-{ϵ}^2}]$, 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 ${\beta }^{\prime }<c<\beta$.

For comparable densities and a fixed mode index, $\eta =({\rho }^{\prime }/\rho ){ϵ}^2\ll 1$. Write $y={\chi }_n-\delta$ near the strong-dispersion transition. In the region controlling the group-velocity minimum of a high-contrast Love wave, $\delta \ll 1$ and $\eta /\delta \gg ϵ$. The leading derivatives giveTheir minimum is atThe frequency estimate follows by inserting ${\delta }_{\min }$ into $W\simeq ({\chi }_n-\delta )[1+{\eta }^2/(2{\delta }^2)+{ϵ}^2/2]$: 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 $K_R^2=W^2-{\chi }_n^2\simeq 2{\chi }_n\Delta$. The finite-load phase correction is $\delta \simeq \eta {\chi }_n/(\kappa h)$; its changes to both $K$ and $dK/dW$ are small ifThe 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 ${\beta }^{\prime }$. 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 ${\mu }^{\prime }/\mu$ is not small. For such materials the exact load criterion $\mu \kappa \gg {\mu }^{\prime }q$ and its frequency derivatives should be used; the preceding contrast estimates assume a bounded order-one density ratio and fixed $n$.