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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.