Past exam of the mathematics course of the University of Cambridge / 2020 / ib / Paper 2 / 16C (Fluid Dynamics)

Equation (1) is Laplace equation for the velocity potential. It expresses incompressibility, $\nabla \cdot \mathbf{u}=0$, together with $\mathbf{u}=\nabla \varphi$. Equation (2) is the linearized dynamic free-surface condition: the unsteady Bernoulli equation and constant atmospheric pressure balance the potential acceleration against the gravitational restoring term $g\zeta$. Equation (3), with the printed $c$ understood as $\zeta$, is the kinematic free-surface condition: the surface moves with the vertical fluid velocity. Equations (4) and (5) are no-penetration boundary conditions at the rigid bottom and cylindrical sidewall, respectively.

For an axisymmetric separated amplitude $\hat{\varphi }(r,z)=R(r)Z(z)$, Laplace equation in cylindrical coordinates becomesThe radial equation isIts solution regular at the axis is the Bessel function $J_0(kr)$. The vertical equation is $Z^{\prime \prime }-k^{2}Z=0$, and the bottom condition $Z^{\prime }(-h)=0$ selectsup to an irrelevant constant multiplier. Finally, the sidewall condition givesTherefore the allowed positive wavenumbers areand

At $z=0$, the two free-surface conditions for a normal mode giveEliminating $\hat{\zeta }$ yieldsFor the vertical factor found in part (b),Thus the gravity-wave dispersion relation iswhere