Past exam of the theoretical mechanics course of the University of Oxford / 2010 / B / 5 / b / i / Solution

Each perturbation potential is harmonic in its own half-space:Write $D_1={\partial }_t+(U/2){\partial }_x$ and $D_2={\partial }_t-(U/2){\partial }_x$. Linearizing the material-interface condition at $z=0$ givesBernoulli's equation at the displaced interface gives $p_i^{\prime }=-{\rho }_i(D_i{\tilde{\varphi }}_i+g\eta )$. Using the supplied pressure jump therefore yieldsThe potentials decay toward their respective infinities. The hydrostatic displacement terms are retained in this dynamic condition.