(a) Assuming a slowly-varying two-dimensional wave pattern of the form

$$\varphi(\mathbf{x}, t)=A(\mathbf{x}, t ; \varepsilon) \exp \left[\frac{i}{\varepsilon} \theta(\mathbf{x}, t)\right]$$

where $0<\varepsilon \ll 1$, and a local dispersion relation $\omega=\Omega(\mathbf{k} ; \mathbf{x}, t)$, derive the ray tracing equations,

$$\frac{d x_{i}}{d t}=\frac{\partial \Omega}{\partial k_{i}}, \quad \frac{d \omega}{d t}=\frac{\partial \Omega}{\partial t}, \quad \frac{d k_{i}}{d t}=-\frac{\partial \Omega}{\partial x_{i}}, \quad \frac{1}{\varepsilon} \frac{d \theta}{d t}=-\omega+k_{j} \frac{\partial \Omega}{\partial k_{j}}$$

for $i, j=1,2$, explaining carefully the meaning of the notation used.

(b) For a homogeneous, time-independent (but not necessarily isotropic) medium, show that all rays are straight lines. When the waves have zero frequency, deduce that if the point $\mathbf{x}$ lies on a ray emanating from the origin in the direction given by a unit vector $\widehat{\mathbf{c}}_{\mathrm{g}}$, then

$$\theta(\mathbf{x})=\theta(\mathbf{0})+\widehat{\mathbf{c}}_{\mathbf{g}} \cdot \mathbf{k}|\mathbf{x}|$$

(c) Consider a stationary obstacle in a steadily moving homogeneous medium which has the dispersion relation

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

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

$$\kappa=\frac{\alpha^{2}}{V^{2} \cos ^{2} \phi}, \quad \widehat{\mathbf{c}}_{\mathrm{g}}=(\cos \psi, \sin \psi)$$

where $\psi$ is defined by

$$\tan \psi=-\frac{\tan \phi}{1+2 \tan ^{2} \phi}$$

with $\frac{1}{2} \pi<\psi<\frac{3}{2} \pi$ and $-\frac{1}{2} \pi<\phi<\frac{1}{2} \pi$. Deduce that the wave pattern occupies a wedge of semi-angle $\tan ^{-1}\left(2^{-3 / 2}\right)$, extending in the negative $x_{1}$-direction.