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

Use the two-dimensional streamfunction convention $u={\psi }_y$, $v=-{\psi }_x$. HereThe divergence is $u_x+v_y=-\cos x\sin y+\cos x\sin y=0$, so the flow is incompressible.

The vorticity is perpendicular to the plane:It is generally nonzero, so this is a rotational flow even though it has a streamfunction.

For steady inviscid flow of constant density without a body-force potential, Bernoulli's theorem givesThe constant can differ between streamlines, because this flow has nonzero vorticity. Dotting the steady Euler equation with $\mathbf{u}$ gives $\mathbf{u}\cdot \nabla (p/\rho +|\mathbf{u}{|}^2/2)=0$, which proves precisely the along-streamline form.

Direct differentiation gives the convective accelerationThe steady Euler equations therefore require $p_x=-\rho \sin x\cos x$ and $p_y=\rho \sin y\cos y$. Integrating both derivatives consistently givesThe additive constant is arbitrary. With a corresponding redefinition of this constant, the Bernoulli function is $p/\rho +|\mathbf{u}{|}^2/2=C_0-{\psi }^2$, making its constancy along streamlines explicit.

Condition (1b) is the kinematic boundary condition: surface particles remain on the moving free-surface graph. Condition (1c) is the dynamic boundary condition for an inviscid interface from unsteady Bernoulli's equation, with the surface pressure fixed at the atmospheric value and absorbed into the potential's time gauge. Condition (1d) is impermeability of the fixed horizontal bed, so its normal velocity vanishes.

Take the undisturbed surface to be $y=0$ and treat $\eta$, $\varphi$, and their induced velocities as first order in wave amplitude. Products of first-order quantities are second order and are discarded. Evaluating surface conditions at $y=\eta$ differs from evaluation at zero only by second-order terms. Thus the linearized surface gravity wave problem isHere $u={\varphi }_x$, $v={\varphi }_y$. The linear free-surface conditions can also be combined into ${\varphi }_{tt}+g{\varphi }_y=0$ at zero.

For $k>0$, phase velocity is the speed of a fixed phase, $c_p=\omega /k$. The group velocity $c_g=d\omega /dk$ transports a narrow-band wave envelope. With $q=kH$,As $H\to 0$ with $k$ fixed, $\tanh (kH)/(kH)\to 1$. ThereforeBoth velocities tend to the shallow-water speed $\sqrt{gH}$ at leading order.