Past exam of the theoretical mechanics course of the University of Oxford / 2009 / B / 4 / c / Solution

At leading order the two wall conditions may be evaluated on the undeformed line $y=b$: shifting the evaluation point multiplies small perturbations and creates only second-order errors. The dynamic relation and the linear Bernoulli relation give $\eta =p^{\prime }/({\rho }_{0}k)=-{\varphi }_t/k$. Differentiate this in time and use ${\varphi }_y={\eta }_t$:The sign follows from the convention that positive pressure displaces the upper wall upward.