= Compressional-wave reflection from a rigid elastic boundary
{title2=$R_P=\sin(\alpha-\beta)/\sin(\alpha+\beta),\quad R_S=-\sin2\alpha/\sin(\alpha+\beta)$}
In an isotropic half-space $y<0$ with zero <displacement> at $y=0$, an incident <longitudinal wave> directed along $(\cos\alpha,\sin\alpha)$ reflects into both longitudinal and <shear waves>. Let $c_p^2=(\lambda+2\mu)/\rho$ and $c_s^2=\mu/\rho$. Horizontal <phase> matching gives $\sin\beta=(c_s/c_p)\cos\alpha$ for a reflected shear <wavevector> proportional to $(\sin\beta,-\cos\beta)$ and <polarization> $(\cos\beta,\sin\beta)$. The two zero-<displacement> equations at the boundary give the displayed coefficients. Since $c_s<c_p$ in a stable isotropic solid, both reflected branches are propagating for every genuine incident angle $0<\alpha<\pi$, with no critical-angle obstruction for this incident-wave type. Changing to a <stress>-free boundary changes the amplitude equations and coefficients.
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