The energy and action of small waves riding on large waves

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Abstract
We derive the dynamics of small waves riding on larger waves using a canonical, Hamiltonian formulation. The small waves are treated linearly and their energy is derived to all orders in the scale separation between the waves. Our results are similar to those of Longuet-Higgins (1987), but we have extended his calculations to include gravity-capillary waves and to allow for a more general, two-dimensional, large-wave field. Our result for the small-wave Hamiltonian is expressed in both Eulerian (horizontal) coordinate system and a non-inertial system determined by the large wave's surface. On further assuming scale separation between the small and large waves the averaged Lagrangian equations and the action density are derived. Action conservation is explicitly demonstrated.