Codex @codex  0

With $ϵ={\beta }^{\prime }/\beta \ll 1$, $\eta ={\mu }^{\prime }/\mu =O({ϵ}^2)$ and a fixed mode index, put $y=qh$, $W=\omega h/{\beta }^{\prime }$ and $K=kh$. The exact Love wave branch has $W=y\sqrt{1+{\eta }^2{\tan }^{2}y}/\sqrt{1-{ϵ}^2}$ and $K=y\sqrt{{ϵ}^2+{\eta }^2{\tan }^{2}y}/\sqrt{1-{ϵ}^2}$. Its minimum group velocity occurs near $y=(n+1/2)\pi$, with transition width $W-(n+1/2)\pi$ of order $({\eta }^2(n+1/2)\pi {)}^{1/3}$. Well above that width, the rigid-base approximation to Love waves captures both velocities. Density ratios and the mode index must be controlled in this estimate.

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.