Waves propagating in a slowly-varying medium satisfy the local dispersion relation $\omega=\Omega(\mathbf{k} ; \mathbf{x}, t)$ in the standard notation. Derive the ray-tracing equations

$$\frac{d x_{i}}{d t}=\frac{\partial \Omega}{\partial k_{i}}, \quad \frac{d k_{i}}{d t}=-\frac{\partial \Omega}{\partial x_{i}}, \quad \frac{d \omega}{d t}=\frac{\partial \Omega}{\partial t}$$

governing the evolution of a wave packet specified by $\varphi(\mathbf{x}, t)=A(\mathbf{x}, t ; \varepsilon) e^{i \theta(\mathbf{x}, t) / \varepsilon}$, where $0<\varepsilon \ll 1$. A formal justification is not required, but the meaning of the $d / d t$ notation should be carefully explained.

The dispersion relation for two-dimensional, small amplitude, internal waves of wavenumber $\mathbf{k}=(k, 0, m)$, relative to Cartesian coordinates $(x, y, z)$ with $z$ vertical, propagating in an inviscid, incompressible, stratified fluid that would otherwise be at rest, is given by

$$\omega^{2}=\frac{N^{2} k^{2}}{k^{2}+m^{2}},$$

where $N$ is the Brunt-Väisälä frequency and where you may assume that $k>0$ and $\omega>0$. Derive the modified dispersion relation if the fluid is not at rest, and instead has a slowly-varying mean flow $(U(z), 0,0)$.

In the case that $U^{\prime}(z)>0, U(0)=0$ and $N$ is constant, show that a disturbance with wavenumber $\mathbf{k}=(k, 0,0)$ generated at $z=0$ will propagate upwards but cannot go higher than a critical level $z=z_{c}$, where $U\left(z_{c}\right)$ is equal to the apparent wave speed in the $x$-direction. Find expressions for the vertical wave number $m$ as $z \rightarrow z_{c}$ from below, and show that it takes an infinite time for the wave to reach the critical level.