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 di-
mensionless version of these equations that is well adapted to the qualitative de-
scription 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 iden-
tify 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 physi-
cal 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.
As everywhere throughout this book, d = 1,2 denotes the spacial dimension of
the surface of the fluid. The spatial variable is written X ∈
and the vertical
variable is denoted by z. We also write X = (x, y) when d = 2 and X = x when
d = 1.
1.1. Mathematical formulation
1.1.1. Basic assumptions. The water waves problem consists in describing
the motion, under the influence of gravity, of a fluid occupying a domain delimited
below by a fixed bottom and above by a free surface that separates it from vacuum
(that is, from a fluid whose density is considered negligible, such as for the air-water
interface). The following assumptions are made on the fluid and the flow:
(H1) The fluid is homogeneous and inviscid.
(H2) The fluid is incompressible.
(H3) The flow is irrotational.
(H4) The surface and the bottom can be parametrized as graphs above the still
(H5) The fluid particles do not cross the bottom.
(H6) The fluid particles do not cross the surface.