= 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 <wave amplitudes> and upward phase-line angles measured from the vertical $\theta_1,\theta_2$, define $r=\widehat\omega_2/\widehat\omega_1$. Then the transmission coefficient is
$$
\frac{A_t}{A_i}=\frac2{r^{-1}\cos\theta_2/\cos\theta_1+r\sin\theta_2/\sin\theta_1}.
$$
The displacement condition is $A_i+A_r=A_t\cos\theta_2/(r\cos\theta_1)$, while pressure continuity gives $A_i-A_r=rA_t\sin\theta_2/\sin\theta_1$. Adding eliminates the reflected amplitude and gives the coefficient. It assumes nonzero intrinsic frequencies and propagating outgoing branches.
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