Improved linear representation of surface waves. Part 2. Slowly varying bottoms and currents

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Abstract
We extend the results of a previous paper to fluids of finite depth. We consider the Hamiltonian theory of waves on the free surface of an incompressible fluid, and derive the canonical transformation that eliminates the leading order of nonlinearity for finite depth. As in the previous paper we propose using the Lie transformation method since it seems to include a nearly correct implementation of short waves interacting with long waves. We show how to use the Eikonal method for slowly varying currents and/or depths in combination with the nonlinear transformation. We note that nonlinear effects are more important in water of finite depth. We note that a nonlinear action conservation law can be derived.