Love-wave cutoff frequencies ($|{\omega }_n|=n\pi /[h\sqrt{{\beta }_0^{-2}-{\beta }^{-2}}]$)

For a slow uniform layer over a faster half-space, the nth Love wave branch reaches the half-space shear wave speed at the displayed threshold. Its half-space decay exponent then vanishes, so the threshold is a limiting nonlocalized state, not a trapped finite-energy mode. The fundamental has zero threshold and exists at every positive frequency; overtones require frequency strictly above their positive threshold. At high frequency each fixed branch approaches the layer shear wave speed from above.