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 velocity amplitudes and upward phase-line angles ${\theta }_1,{\theta }_2$, define $r={\hat{\omega }}_2/{\hat{\omega }}_1$. Then the incident-to-transmitted amplitude ratio isThis follows by adding the displacement and pressure matching equations after eliminating the reflected amplitude. It assumes nonzero intrinsic frequencies and propagating outgoing branches.