ABSTRACT
Consider two-dimensional progressive gravity waves moving at a phase speed C on the surface of infinitely deep water. The problem is defined in a coordinate system (x, y) fixed to the waves, with the x-axis positive in the direction of wave propagation and the y-axis pointing vertically upward from the stillwater level. Assume that the fluid is inviscid, incompressible, and without surface tension. Let φ(x, y) denote the velocity potential and ζ(x) the wave elevation, respectively. The fluid motion can be described by the Laplace equation
∇2φ(x, y) = 0 for (x, y) ∈ Ω, (18.1) where
Ω = {(x, y) | −∞ < x < +∞,−∞ < y < ζ(x)} . The velocity potential φ(x, y) is subject to the free surface boundary conditions
C2φxx + gφy + 1 2 ∇φ∇(∇φ∇φ)− 2C∇φ∇φx = 0 at y = ζ(x), (18.2)
ζ(x) = 1 g
( Cφx − 12∇φ∇φ
) at y = ζ(x), (18.3)
and the bottom condition
lim y→−∞
∂φ
∂y = 0, (18.4)
where g is the acceleration of gravity and the subscripts x and y denote partial derivatives in the respective directions.
