Derive the ray-tracing equations governing the evolution of a wave packet $\phi(\mathbf{x}, t)=$ $A(\mathbf{x}, t) \exp \{i \psi(\mathbf{x}, t)\}$ in a slowly varying medium, stating the conditions under which the equations are valid.

Consider now a stationary obstacle in a steadily moving homogeneous two-dimensional medium which has the dispersion relation

$$\omega\left(k_{1}, k_{2}\right)=\alpha\left(k_{1}^{2}+k_{2}^{2}\right)^{1 / 4}-V k_{1}$$

where $(V, 0)$ is the velocity of the medium. The obstacle generates a steady wave system. Writing $\left(k_{1}, k_{2}\right)=\kappa(\cos \phi, \sin \phi)$, show that the wave satisfies

$$\kappa=\frac{\alpha^{2}}{V^{2} \cos ^{2} \phi}$$

Show that the group velocity of these waves can be expressed as

$$\mathbf{c}_{g}=V\left(\frac{1}{2} \cos ^{2} \phi-1, \frac{1}{2} \cos \phi \sin \phi\right) .$$

Deduce that the waves occupy a wedge of semi-angle $\sin ^{-1} \frac{1}{3}$ about the negative $x_{1}$-axis.