The dispersion relation for sound waves of frequency $\omega$ in a stationary homogeneous gas is $\omega=c_{0}|\mathbf{k}|$, where $c_{0}$ is the speed of sound and $\mathbf{k}$ is the wavenumber. Derive the dispersion relation for sound waves of frequency $\omega$ in a uniform flow with velocity U.

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

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

explaining carefully the meaning of the notation used.

Suppose that two-dimensional sound waves with initial wavenumber $\left(k_{0}, l_{0}, 0\right)$ are generated at the origin in a gas occupying the half-space $y>0$. If the gas has a slowlyvarying mean velocity $(\gamma y, 0,0)$, where $\gamma>0$, show:

(a) that if $k_{0}>0$ and $l_{0}>0$ the waves reach a maximum height (which should be identified), and then return to the level $y=0$ in a finite time;

(b) that if $k_{0}<0$ and $l_{0}>0$ then there is no bound on the height to which the waves propagate.

Comment briefly on the existence, or otherwise, of a quiet zone.