The dispersion relation for sound waves of frequency $\omega$ in a stationary, homogeneous gas is $\omega=c|\mathbf{k}|$, where $c$ is the speed of sound and $\mathbf{k}$ is the wavevector. Derive the dispersion relation for sound waves of frequency $\omega$ in a uniform flow with velocity U.

For a slowly-varying medium with a local dispersion relation $\omega=\Omega(\mathbf{k} ; \mathbf{x}, t)$, derive the ray-tracing equations

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

The meaning of the notation $d / d t$ should be carefully explained.

Suppose that two-dimensional sound waves with initial wavevector $\left(k_{0}, l_{0}\right)$ are generated at the origin in a gas occupying the half-space $y>0$. The gas has a mean velocity $(\gamma y, 0)$, where $0<\gamma \ll\left(k_{0}^{2}+l_{0}^{2}\right)^{\frac{1}{2}}$. Show that

(a) if $k_{0}>0$ and $l_{0}>0$ then an initially upward propagating wavepacket returns to the level $y=0$ within a finite time, after having reached a maximum height that should be identified;

(b) if $k_{0}<0$ and $l_{0}>0$ then an initially upward propagating wavepacket continues to propagate upwards for all time.

For the case of a fixed frequency disturbance comment briefly on whether or not there is a quiet zone.