The so-called "shallow water theory" is characterised by the equations

$$\begin{aligned} &\frac{\partial \zeta}{\partial t}+\frac{\partial}{\partial x}[(h+\zeta) u]=0 \\ &\frac{\partial u}{\partial t}+u \frac{\partial u}{\partial x}+g \frac{\partial \zeta}{\partial x}=0 \end{aligned}$$

where $g$ denotes the gravitational constant, the constant $h$ denotes the undisturbed depth of the water, $u(x, t)$ denotes the speed in the $x$-direction, and $\zeta(x, t)$ denotes the elevation of the water.

(i) Assuming that $|u|$ and $|\zeta|$ and their gradients are small in some appropriate dimensional considerations, show that $\zeta$ satisfies the wave equation

$$\frac{\partial^{2} \zeta}{\partial t^{2}}=c^{2} \frac{\partial^{2} \zeta}{\partial x^{2}},$$

where the constant $c$ should be determined in terms of $h$ and $g$.

(ii) Using the change of variables

$$\xi=x+c t, \quad \eta=x-c t$$

show that the general solution of $(*)$ satisfying the initial conditions

$$\zeta(x, 0)=u_{0}(x), \quad \frac{\partial \zeta}{\partial t}(x, 0)=v_{0}(x)$$

is given by

$$\zeta(x, t)=f(x+c t)+g(x-c t)$$

where

$$\begin{aligned} &\frac{d f(x)}{d x}=\frac{1}{2}\left[\frac{d u_{0}(x)}{d x}+\frac{1}{c} v_{0}(x)\right] \\ &\frac{d g(x)}{d x}=\frac{1}{2}\left[\frac{d u_{0}(x)}{d x}-\frac{1}{c} v_{0}(x)\right] \end{aligned}$$

Simplify the above to find $\zeta$ in terms of $u_{0}$ and $v_{0}$.

(iii) Find $\zeta(x, t)$ in the particular case that

$$u_{0}(x)=H(x+1)-H(x-1), \quad v_{0}(x)=0, \quad-\infty<x<\infty$$

where $H(\cdot)$ denotes the Heaviside step function.

Describe in words this solution.