= Solution
Equation (1) is <Laplace equation> for the <velocity potential>. It expresses incompressibility, $\nabla\cdot\mathbf u=0$, together with $\mathbf u=\nabla\phi$. 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.
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