Past exam of the options I course of the University of Oxford / 2008 / K (Fluid Dynamics and Waves)

Use the stream function convention $u={\psi }_y$, $v=-{\psi }_x$. It gives $u=-\sin x\sin y$, $v=-\cos x\cos y$. Thenso the flow is incompressible.

The vorticity points normally to the plane and has scalar componentIt is nonzero in general, so the flow is rotational.

For this steady, inviscid, constant-density flow without a body force, Bernoulli's theorem says thatIts constant may vary from one streamline to another because the flow has vorticity.

The convective acceleration isSteady Euler equations therefore give $p_x=-\rho \sin x\cos x$ and $p_y=\rho \sin y\cos y$. Integrating these compatible equations yieldswith arbitrary additive constant $p_0$. As a check, $p/\rho +(u^2+v^2)/2=p_0/\rho +1/2-{\psi }^2$, indeed constant on each streamline. This is a Steady Euler flow with vorticity proportional to its stream function.

Condition (1b) is the kinematic free-surface boundary condition: fluid particles on the surface remain there, since the material derivative of $y-\eta (x,t)$ vanishes. Condition (1c) is the dynamic free-surface boundary condition, obtained from the unsteady Bernoulli equation with constant atmospheric pressure and a choice of potential gauge. Condition (1d) is impermeability: no normal flow through the horizontal bed.

Write the surface displacement and the perturbation velocity potential as small quantities about rest, and drop their quadratic products. Evaluating the remaining surface terms at the unperturbed level $y=0$ givesThe velocity components are $u={\varphi }_x$, $v={\varphi }_y$. The shift of the evaluation point from zero to $\eta$ contributes a product of perturbations, as do $u{\eta }_x$ and the kinetic-energy term, so they disappear at this order.

For positive wavenumber $k$, the phase velocity is $c_p=\omega /k$, the speed of a fixed phase. The group velocity is $c_g=d\omega /dk$, the leading speed of a narrow wave packet. Differentiating the supplied dispersion relation givesFor fixed $k>0$, the shallow-depth limit uses $\tanh s/s\to 1$: