Rigid-base approximation to Love waves (${\omega }_n={\beta }^{\prime }(n+\frac{1}{2})\pi /h$)

A free top and rigid base impose transverse wavenumbers $(n+1/2)\pi /h$ on guided SH modes between rigid and free planes. Their phase velocity and group velocity are $c={\beta }^{\prime }/\sqrt{1-{\omega }_n^2/{\omega }^2}$ and $U={\beta }^{\prime }\sqrt{1-{\omega }_n^2/{\omega }^2}$ for $\omega >{\omega }_n$. A real Love wave replaces the rigid base by an evanescent elastic load; the approximation becomes reliable sufficiently above the corresponding strong-dispersion transition, not at the actual mode cutoff.

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.