Rayleigh wave decay exponents ($b_L=k\sqrt{1-c^2/c_L^2},b_T=k\sqrt{1-c^2/c_T^2}$)

In a homogeneous isotropic elastic half-space $Y>0$, a Rayleigh wave with factor $e^{ik(X-ct)}$ has two depth factors $e^{-b_{L}Y}$ and $e^{-b_{T}Y}$. Substitution into the Navier-Cauchy equation factors the amplitude determinant into longitudinal and transverse factors, giving the displayed exponents. Strict exponential decay requires $0<c<\min (c_L,c_T)$. The longitudinal constituent has $(A,B)\propto (ik,-b_L)$; the transverse constituent has $ikA-b_{T}B=0$. The free-surface traction conditions select a combination of both constituents.