The Water Waves Problem and Its Asymptotic Regimes

We derive here various equivalent mathematical formulations of the water waves problem (and some extensions to the two-fluids problem). We then propose a dimensionless version of these equations that is well adapted to the qualitative description of the solutions. The way we nondimensionalize the water waves equations relies on a rough analysis of their linearization around the rest state and shows the relevance of various dimensionless parameters, namely, the amplitude parameter ε, the shallowness parameter μ, the topography parameter β, and the transversality parameter γ. The linear analysis of the equations is also used to introduce the concept of wave packets and modulation equations. With the relevant physical dimensionless parameters introduced, we then identify asymptotic regimes (the shallow water regime for instance) as conditions on these dimensionless parameters (e.g., μ 1 for the shallow water regime). Finally, we present two natural extensions of the problems addressed in this book: the case of moving bottoms and of rough topographies. A discussion of the main physical assumptions (e.g., homogeneity, inviscidity, incompressibility, etc.) and some comments on possible extensions (such as taking into account Coriolis effects, or a nonconstant external pressure) are then briefly addressed in the last section.