Write down expressions for the phase speed $c$ and group velocity $c_{g}$ in one dimension for general waves of the form $A \exp [i(k x-\omega t)]$ with dispersion relation $\omega(k)$. Briefly indicate the physical significance of $c$ and $c_{g}$ for a wavetrain of finite size.

The dispersion relation for internal gravity waves with wavenumber $\mathbf{k}=(k, 0, m)$ in an incompressible stratified fluid with constant buoyancy frequency $N$ is

$$\omega=\frac{\pm N k}{\left(k^{2}+m^{2}\right)^{1 / 2}} .$$

Calculate the group velocity $\mathbf{c}_{g}$ and show that it is perpendicular to $\mathbf{k}$. Show further that the horizontal components of $\mathbf{k} / \omega$ and $\mathbf{c}_{g}$ have the same sign and that the vertical components have the opposite sign.

The vertical velocity $w$ of small-amplitude internal gravity waves is governed by

$$\frac{\partial^{2}}{\partial t^{2}}\left(\nabla^{2} w\right)+N^{2} \nabla_{h}^{2} w=0$$

where $\nabla_{h}^{2}$ is the horizontal part of the Laplacian and $N$ is constant.

Find separable solutions to $(*)$ of the form $w(x, z, t)=X(x-U t) Z(z)$ corresponding to waves with constant horizontal phase speed $U>0$. Comment on the nature of these solutions for $0<k<N / U$ and for $k>N / U$.

A semi-infinite stratified fluid occupies the region $z>h(x, t)$ above a moving lower boundary $z=h(x, t)$. Construct the solution to $(*)$ for the case $h=\epsilon \sin [k(x-U t)]$, where $\epsilon$ and $k$ are constants and $\epsilon \ll 1$.

Sketch the orientation of the wavecrests, the propagation direction and the group velocity for the case $0<k<N / U$.

Part II