The dispersion relation for capillary waves on the surface of deep water is

$$\omega^{2}=S^{2}|k|^{3}$$

where $S=(T / \rho)^{1 / 2}, \rho$ is the density and $T$ is the coefficient of surface tension. The free surface $z=\eta(x, t)$ is undisturbed for $t<0$, when it is suddenly impacted by an object, giving the initial conditions at time $t=0$ :

$$\eta=0 \quad \text { and } \quad \frac{\partial \eta}{\partial t}= \begin{cases}-W, & |x|<\epsilon \\ 0, & |x|>\epsilon\end{cases}$$

where $W$ is a constant.

(i) Use Fourier analysis to find an integral expression for $\eta(x, t)$ when $t>0$.

(ii) Use the method of stationary phase to find the asymptotic behaviour of $\eta(V t, t)$ for fixed $V>0$ as $t \rightarrow \infty$, for the case $V \ll \epsilon^{-1 / 2} S$. Show that the result can be written in the form

$$\eta(x, t) \sim \frac{W \epsilon S t^{2}}{x^{5 / 2}} F\left(\frac{x^{3}}{S^{2} t^{2}}\right)$$

and determine the function $F$.

(iii) Give a brief physical interpretation of the link between the condition $\epsilon V^{2} / S^{2} \ll$ 1 and the simple dependence on the product $W \epsilon$.

[You are given that $\int_{-\infty}^{\infty} e^{\pm i a u^{2}} d u=(\pi / a)^{1 / 2} e^{\pm i \pi / 4} \quad$ for $a>0 .$ ]