Paper 3, Section II, B

Write down the exact kinematic and dynamic boundary conditions that apply at the free surface $z=\eta(x, t)$ of a fluid layer in the presence of gravity in the $z$-direction. Show how these may be approximated for small disturbances of a hydrostatic state about $z=0$. (The flow of the fluid is in the $(x, z)$-plane and may be taken to be irrotational, and the pressure at the free surface may be assumed to be constant.)

Fluid of density $\rho$ fills the region $0>z>-h$. At $z=-h$ the $z$-component of the velocity is $\epsilon \operatorname{Re}\left(e^{i \omega t} \cos k x\right)$, where $|\epsilon| \ll 1$. Find the resulting disturbance of the free surface, assuming this to be small. Explain physically why your answer has a singularity for a particular value of $\omega^{2}$.

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