Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 78 / 3

Let the layer occupy $0<z<h$, with a free surface at $z=0$ and rigid base at $z=h$. Write an SH-wave as $u_y=Y(z)e^{i(kx-\omega t)}$. The equation and boundary conditions giveThus $Y=A\cos (qz)$ and the guided SH modes between rigid and free planes haveThe propagating branch has $\omega >{\omega }_n$ and $k=\sqrt{{\omega }^2-{\omega }_n^2}/{\beta }^{\prime }$. Hence the rigid-base approximation to Love waves predictsAt each threshold $c_n\to \infty$ and $U_n\to 0$. As frequency rises, the phase velocity decreases towards ${\beta }^{\prime }$ and the group velocity increases towards ${\beta }^{\prime }$. Frequencies below ${\omega }_n$ have imaginary horizontal wavenumber and do not give propagating guided modes. The infinite threshold phase speed does not violate causality: the limiting group speed is zero. These quarter-wave thresholds differ from the actual Love-wave cutoff frequencies of an elastic substrate.

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 $n$.

With vertical shear modulus ${\rho }^{\prime }{\beta }_z^{\prime 2}$ and horizontal shear modulus ${\rho }^{\prime }{\beta }_x^{\prime 2}$, force balance therefore gives the anisotropic shear-horizontal wave equationFor $u_y=Y(z)e^{i(kx-\omega t)}$, the layer vertical wavenumber and substrate decay exponent areThe free top still gives $Y=A\cos (qz)$; the substrate still has $Y=B{e}^{-\kappa (z-h)}$. Welded displacement continuity and the now anisotropic traction continuity giveHence the anisotropic Love-wave dispersion relation isEquivalently, writing $c=\omega /k$,This replaces the isotropic speed in the layer equation by separate horizontal and vertical speeds and replaces the layer traction modulus by ${\rho }^{\prime }{\beta }_z^{\prime 2}$. No additional SH/P-SV coupling arises for the aligned symmetry axes and constitutive law here.

A rigid base still imposes $q_n={\chi }_n/h$, but nowThusThe vertical speed sets the resonance frequencies, while the horizontal speed sets the high-frequency limits and $c_n{U}_n={\beta }_x^{\prime 2}$.

For the elastic substrate, put ${ϵ}_x={\beta }_x^{\prime }/\beta$, $\zeta ={\rho }^{\prime }{\beta }_x^{\prime }{\beta }_z^{\prime }/(\rho {\beta }^2)$, $W=\omega h/{\beta }_z^{\prime }$ and $K=kh{\beta }_x^{\prime }/{\beta }_z^{\prime }$. The exact parameterization isIt reproduces the isotropic analysis with ${\beta }^{\prime }$ replaced by ${\beta }_z^{\prime }$ in the frequency scale, by ${\beta }_x^{\prime }$ in the velocities, and by the two effective contrast parameters above. In particular the true cutoffs are ${\omega }_{c,n}=n\pi {\beta }_z^{\prime }/[h\sqrt{1-{ϵ}_x^2}]$, the group minima occur near ${\omega }_{n,z}$, and a sufficient rigid-limit validity range isFor comparable horizontal and vertical layer speeds, a bounded density ratio and fixed mode index, the minimum estimate becomes ${\delta }_{\min }\sim ({\zeta }^2{\chi }_n/2{)}^{1/3}$ and $U_{\min }/{\beta }_x^{\prime }\sim 3(\zeta /{\chi }_n{)}^{1/3}/2^{2/3}$. If their ratio is extreme, that more specific minimum-depth estimate additionally needs $\zeta /{\delta }_{\min }\gg {ϵ}_x$; the exact parameterization and load criterion remain valid without it. As in the isotropic case, neither a slow horizontal nor a slow vertical speed alone justifies the rigid boundary at cutoff or within the strong-dispersion transition.