= Internal-wave transmission across a velocity jump
Across a horizontal material interface with continuous background <mass density> but a velocity jump, an <internal gravity wave> preserves laboratory <frequency> and horizontal <wavenumber>. The matching conditions are continuity of displacement $w/\widehat\omega$ and <pressure>, not continuity of $w$. For velocity amplitudes and upward phase-line angles $\theta_1,\theta_2$, define $r=\widehat\omega_2/\widehat\omega_1$. Then the incident-to-transmitted amplitude ratio is
$$
\frac{A_t}{A_i}=\frac2{r\cos\theta_2/\cos\theta_1+r^{-1}\sin\theta_2/\sin\theta_1}.
$$
This follows by adding the displacement and pressure matching equations after eliminating the reflected amplitude. It assumes nonzero intrinsic frequencies and propagating outgoing branches.
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