The equations describing small-amplitude motions in a stably stratified, incompressible, inviscid fluid are

$$\frac{\partial \tilde{\rho}}{\partial t}+w \frac{\mathrm{d} \rho_{0}}{\mathrm{~d} z}=0, \quad \rho_{0} \frac{\partial \mathbf{u}}{\partial t}=\tilde{\rho} \mathbf{g}-\nabla \tilde{p}, \quad \boldsymbol{\nabla} \cdot \mathbf{u}=0$$

where $\rho_{0}(z)$ is the background stratification, $\tilde{\rho}(\mathbf{x}, t)$ and $\tilde{p}(\mathbf{x}, t)$ are the perturbations about an undisturbed hydrostatic state, $\mathbf{u}(\mathbf{x}, t)=(u, v, w)$ is the velocity, and $\mathbf{g}=(0,0,-g)$.

Show that

$$\left[\frac{\partial^{2}}{\partial t^{2}} \nabla^{2}+N^{2}\left(\nabla^{2}-\frac{\partial^{2}}{\partial z^{2}}\right)\right] w=0$$

stating any approximation made, and define the Brunt-Väisälä frequency $N$.

Deduce the dispersion relation for plane harmonic waves with wavevector $\mathbf{k}=$ $(k, 0, m)$. Calculate the group velocity and verify that it is perpendicular to $\mathbf{k}$.

Such a stably stratified fluid with a uniform value of $N$ occupies the region $z>h(x, t)$ above a moving lower boundary $z=h(x, t)$. Find the velocity field $w(x, z, t)$ generated by the boundary motion for the case $h=\epsilon \sin [k(x-U t)]$, where $0<\epsilon k \ll 1$ and $U>0$ is a constant.

For the case $k^{2}<N^{2} / U^{2}$, sketch the orientation of the wave crests, the direction of propagation of the crests, and the direction of the group velocity.