Consider a thin circular sheet of water of radius, \(r_{o}\), and depth, \(h\), that is being excited by steady, periodic oscillations of its outer rim at a frequency, \(\omega\). We want to catalog the types of tidal waves that can form.

a. Show that the equation describing the wave height in two dimensions is:

\[\frac{\partial^{2} \delta_{\lambda}}{\partial r^{2}}+\frac{1}{r} \frac{\partial \delta_{\lambda}}{\partial r}+\frac{1}{r^{2}} \frac{\partial^{2} \delta_{\lambda}}{\partial \theta^{2}}+k^{2} \delta_{\lambda}=0 \quad k=\frac{\omega}{\sqrt{g h}}\]

b. Solve this equation in generic form using the boundary conditions:

\[\begin{array}{lll}r=r_{o} & \frac{\partial \delta_{\lambda}}{\partial r}=0 & \text { Wave rigidly follows the outer boundary } \=0 & \delta_{\lambda}=\text { finite } & \text { Wave height must be bounded at origin }\end{array}\]

Show that the following solutions are valid:

\[\delta_{\lambda}=A_{s} J_{s}(k r) \cos (s \theta) \cos (\omega t)+B_{s} J_{s}(k r) \sin (s \theta) \cos (\omega t)\]

What are the allowable values of \(k\) ?

c. Each value of \(s\) in the solutions above yield a different mode of wave. Plot several streamlines for the first two \(s=0\) and \(s=1\) modes of the two families of solutions and note the location of the nodes.