 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 $w/\hat{\omega }$ and pressure, not continuity of $w$. For wave amplitudes and upward phase-line angles measured from the vertical ${\theta }_1,{\theta }_2$, define $r={\hat{\omega }}_2/{\hat{\omega }}_1$. Then the transmission coefficient isThe displacement condition is $A_i+A_r=A_t\cos {\theta }_2/(r\cos {\theta }_1)$, while pressure continuity gives $A_i-A_r=r{A}_t\sin {\theta }_2/\sin {\theta }_1$. Adding eliminates the reflected amplitude and gives the coefficient. It assumes nonzero intrinsic frequencies and propagating outgoing branches.