Past exam of the options I course of the University of Oxford / 2010 / K (Fluid Dynamics and Waves)

For a material volume $V(t)$, Newton's second law equates the rate of change of momentum to pressure and body forces:Apply the supplied Reynolds transport theorem to each component of $\mathbf{u}$, and apply the divergence theorem to the pressure force. The result isSince the material volume at the chosen time is arbitrary, the integrand vanishes pointwise for smooth fields. Dividing by positive density gives the Euler equationsAt a fixed time $t$, a streamline is a curve tangent everywhere to the instantaneous velocity. Parametrize it by $d\mathbf{x}/ds=\mathbf{u}(\mathbf{x}(s),t)$. Thenso $f$ is constant along it when the displayed directional derivative vanishes. Here time is held fixed: streamlines need not coincide with particle paths in an unsteady flow.

For $k>0$, tryThe potential satisfies Laplace's equation, and its derivative with respect to $y$ vanishes at $y=-h$. The surface kinematic condition givesThe dynamic condition gives $-B\omega \cosh (kh)+gA=0$. Eliminating the nonzero amplitudes yieldsThus choosing either frequency sign satisfying this relation and $B=A\omega /(k\sinh (kh))$ produces the required harmonic wave.