Starting from the equations of motion for an inviscid, incompressible, stratified fluid of density $\rho_{0}(z)$, where $z$ is the vertical coordinate, derive the dispersion relation

$$\omega^{2}=\frac{N^{2}\left(k^{2}+\ell^{2}\right)}{\left(k^{2}+\ell^{2}+m^{2}\right)}$$

for small amplitude internal waves of wavenumber $(k, \ell, m)$, where $N$ is the constant Brunt-Väisälä frequency (which should be defined), explaining any approximations you make. Describe the wave pattern that would be generated by a small body oscillating about the origin with small amplitude and frequency $\omega$, the fluid being otherwise at rest.

The body continues to oscillate when the fluid has a slowly-varying velocity $[U(z), 0,0]$, where $U^{\prime}(z)>0$. Show that a ray which has wavenumber $\left(k_{0}, 0, m_{0}\right)$ with $m_{0}<0$ at $z=0$ will propagate upwards, but cannot go higher than $z=z_{c}$, where

$$U\left(z_{c}\right)-U(0)=N\left(k_{0}^{2}+m_{0}^{2}\right)^{-1 / 2}$$

Explain what happens to the disturbance as $z$ approaches $z_{c}$.