Past exam of the mathematics course of the University of Cambridge / 2008 / ib / Paper 1 / 17B / Solution

Distinguish the full velocity potentials ${\Phi }_i=U_{i}x+{\phi }_i$ from their perturbations ${\phi }_i$. The linear boundary conditions use the perturbations. This distinction reconciles the notation in the printed conditions with the full potentials written in the mode ansatz. All equations below neglect products of perturbations. Assume positive densities and no surface tension.