Matched asymptotics of Rayleigh-wave fields near cuspidal ridges and gorges

We construct a local matched-asymptotic description of time-harmonic elastic fields generated by Rayleigh waves near cuspidal elements of a traction-free surface. The free surface is represented locally by a cusp graph with exponent $0<\alpha<1$, or equivalently by a vanishing-width horn $b(s)=B s^m$, $m=1/\alpha>1$. A cuspidal gorge is a zero-opening re-entrant notch: its leading field is the Williams crack-tip field, and the stresses behave as $r^{-1/2}$. The cusp exponent affects the gorge through lower-order corrections and through the stress cut-off produced by rounding the bottom. In contrast, a cuspidal ridge behaves as an elastic horn with vanishing width. The leading admissible free-tip field is asymptotically rigid (bounded stress), distinct from the high-energy branch supported by a finite tip truncation, where stresses grow as $\sigma \sim \rho^{-m}$. Finite-element calculations for the local static Lam\'e problems support these predictions: the free-tip ridge test confirms the absence of crack-like growth, the truncated ridge recovers the high-energy law, and the gorge stress slope is found to be close to $-1/2$.