Past exam of the options I course of the University of Oxford / 2013 / K (Fluid Dynamics and Waves)

Use the complex potential of a two-dimensional potential flow convention $W=\varphi +i\psi$, for which $W^{\prime }(z)=u-iv$. A positive circulation is counterclockwise. The two point vortices and the two-dimensional point sink giveBranch choices change additive constants, not the local velocity field. A vortex is transported by the nonsingular velocity due to the other vortex and the sink; its own singular term is omitted. At $z=c$ this givesThe other vortex follows $-c$ by symmetry. Put $c=r{e}^{i\theta }$, with $r(0)=a$ and $\theta (0)=0$. Multiplication by $\bar{c}$ givesFor $Q>0$, integration yieldsEquivalently, with $s=1-Qt/(\pi a^2)$,The curves obey $\theta =(\Gamma /(2Q))\log (a/r)$, so they are logarithmic spirals. At $t=\pi a^2/Q$, both vortices reach the sink at the origin. For nonzero circulation they make infinitely many turns in that finite time; for zero circulation they approach radially. The point-vortex evolution becomes singular at capture, so the formula does not define a regular continuation through the event.

Put $\tau =T/\rho >0$ and take the positive-frequency branch for $k>0$. The phase velocity and group velocity areSince the square root is increasing, minimizing $c_p$ is equivalent to minimizing $g/k+\tau k$. Its derivative is $-g/k^2+\tau$, which changes sign from negative to positive atThe phase speed diverges at both positive endpoints $k\downarrow 0$ and $k\to \infty$, so this stationary point is its global minimum. At $k_c$, $\tau k_c^2=g$, givingThe relative speed is especially transparent:A narrow wave packet moves with $c_g$, while its crests move with $c_p$. For $k<k_c$, $c_p>c_g$: crests move forward through the packet, from its rear towards its front. For $k>k_c$, $c_p<c_g$: crests drift backwards relative to the packet, although both absolute velocities are positive. At $k_c$ the leading-order relative drift vanishes. This is the minimum phase speed of a gravity-capillary wave.