Internal-wave transmission across a buoyancy-frequency jump ($\omega =N_j\cos {\theta }_j$)

At a jump in buoyancy frequency with continuous background density, incident, reflected and transmitted internal gravity waves share frequency and horizontal wavenumber. Continuity of pressure and normal velocity determines their amplitudes. If amplitudes multiply unit particle-displacement vectors directed along each ray with positive horizontal component, the reflection coefficient is $[\sin {\theta }_2/\sin {\theta }_1-\cos {\theta }_2/\cos {\theta }_1]/[\sin {\theta }_2/\sin {\theta }_1+\cos {\theta }_2/\cos {\theta }_1]$. Using vertical-displacement amplitudes reverses the reflected coefficient's sign because the reflected polarization reverses its vertical component.

A reflecting bottom beneath a jump in buoyancy frequency creates a standing field from repeated transmitted and reflected internal gravity waves. For ray angles ${\theta }_1=\pi /3$ and ${\theta }_2=\pi /6$, the downward particle-displacement amplitude has magnitude $\sqrt{3}|{\eta }_i|/[1+8{\sin }^2(m_{2}H){]}^{1/2}$. It reaches $\sqrt{3}|{\eta }_i|$ when $m_{2}H$ is an integer multiple of $\pi$. The upper reflected wave has the same amplitude magnitude as the incident wave because the lossless closed cavity admits no net downward energy flux.