Paper 4, Section II, D

A deep layer of inviscid fluid is initially confined to the region $0<x<a, 0<y<a$, $z<0$ in Cartesian coordinates, with $z$ directed vertically upwards. An irrotational disturbance is caused to the fluid so that its upper surface takes position $z=\eta(x, y, t)$. Determine the linear normal modes of the system and the dispersion relation between the frequencies of the normal modes and their wavenumbers.

If the interface is initially displaced to position $z=\epsilon \cos \frac{3 \pi x}{a} \cos \frac{4 \pi y}{a}$ and released from rest, where $\epsilon$ is a small constant, determine its position for subsequent times. How far below the surface will the velocity have decayed to $1 / e$ times its surface value?

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