Refraction of a stationary internal-wave causal envelope
= Refraction of a stationary internal-wave causal envelope
A lower ray reaches $H$ after $\tau_1=H/[U_1\sin\theta_1\cos\theta_1]$, at $x_H=H\cot\theta_1$. Conserved horizontal <wavenumber> gives $\cos\theta_2=(U_2N_1/U_1N_2)\cos\theta_1$. The transmitted front is $(x_H,H)+(t-\tau_1)\mathbf c_{g2}$ on propagating branches; reflection reverses the vertical <group velocity>.