A vertical cylindrical container of radius $R$ is partly filled with fluid of constant density to depth $h$. The free surface is perturbed so that the fluid occupies the region

$$0<r<R, \quad-h<z<\zeta(r, \theta, t)$$

where $(r, \theta, z)$ are cylindrical coordinates and $\zeta$ is the perturbed height of the free surface. For small perturbations, a linearised description of surface waves in the cylinder yields the following system of equations for $\zeta$ and the velocity potential $\phi$ :

$$\begin{aligned} \nabla^{2} \phi &=0, \quad 0<r<R, \quad-h<z<0 \\ \frac{\partial \phi}{\partial t}+g \zeta &=0 \quad \text { on } \quad z=0 \\ \frac{\partial \zeta}{\partial t}-\frac{\partial \phi}{\partial z} &=0 \quad \text { on } \quad z=0 \\ \frac{\partial \phi}{\partial z} &=0 \quad \text { on } \quad z=-h \\ \frac{\partial \phi}{\partial r} &=0 \quad \text { on } \quad r=R \end{aligned}$$

(a) Describe briefly the physical meaning of each equation.

(b) Consider axisymmetric normal modes of the form

$$\phi=\operatorname{Re}\left(\hat{\phi}(r, z) e^{-i \sigma t}\right), \quad \zeta=\operatorname{Re}\left(\hat{\zeta}(r) e^{-i \sigma t}\right)$$

Show that the system of equations $(1)-(5)$ admits a solution for $\hat{\phi}$ of the form

$$\hat{\phi}(r, z)=A J_{0}\left(k_{n} r\right) Z(z)$$

where $A$ is an arbitrary amplitude, $J_{0}(x)$ satisfies the equation

$$\frac{d^{2} J_{0}}{d x^{2}}+\frac{1}{x} \frac{d J_{0}}{d x}+J_{0}=0$$

the wavenumber $k_{n}, n=1,2, \ldots$ is such that $x_{n}=k_{n} R$ is one of the zeros of the function $d J_{0} / d x$, and the function $Z(z)$ should be determined explicitly.

(c) Show that the frequency $\sigma_{n}$ of the $n$-th mode is given by

$$\sigma_{n}^{2}=\frac{g}{h} \Psi\left(k_{n} h\right)$$

where the function $\Psi(x)$ is to be determined.

[Hint: In cylindrical coordinates $(r, \theta, z)$,

$$\left.\nabla^{2}=\frac{1}{r} \frac{\partial}{\partial r}\left(r \frac{\partial}{\partial r}\right)+\frac{1}{r^{2}} \frac{\partial^{2}}{\partial \theta^{2}}+\frac{\partial^{2}}{\partial z^{2}} \cdot\right]$$