The dispersion relation in a stationary medium is given by $\omega=\Omega_{0}(\mathbf{k})$, where $\Omega_{0}$ is a known function. Show that, in the frame of reference where the medium has a uniform velocity $-\mathbf{U}$, the dispersion relation is given by $\omega=\Omega_{0}(\mathbf{k})-\mathbf{U} \cdot \mathbf{k}$.

An aircraft flies in a straight line with constant speed $M c_{0}$ through air with sound speed $c_{0}$. If $M>1$ show that, in the reference frame of the aircraft, the steady waves lie behind it on a cone of semi-angle $\sin ^{-1}(1 / M)$. Show further that the unsteady waves are confined to the interior of the cone.

A small insect swims with constant velocity $\mathbf{U}=(U, 0)$ over the surface of a pool of water. The resultant capillary waves have dispersion relation $\omega^{2}=T|\mathbf{k}|^{3} / \rho$ on stationary water, where $T$ and $\rho$ are constants. Show that, in the reference frame of the insect, steady waves have group velocity

$$\mathbf{c}_{g}=U\left(\frac{3}{2} \cos ^{2} \beta-1, \frac{3}{2} \cos \beta \sin \beta\right),$$

where $\mathbf{k} \propto(\cos \beta, \sin \beta)$. Deduce that the steady wavefield extends in all directions around the insect.