The gravity wave represents a perturbation around a stationary state, in which there is no velocity. Thus, the perturbation introduced to the system is described by a velocity field of infinitesimally small amplitude,

Because the fluid is assumed incompressible, this velocity field has the

streamfunction representation

where the subscripts indicate

partial derivatives. In this derivation it suffices to work in two dimensions

, where gravity points in the negative

*z*-direction. Next, in an initially stationary incompressible fluid, there is no vorticity, and the fluid stays

irrotational, hence

. In the streamfunction representation,

Next, because of the translational invariance of the system in the

*x*-direction, it is possible to make the

ansatz
where

*k* is a spatial wavenumber. Thus, the problem reduces to solving the equation

We work in a sea of infinite depth, so the boundary condition is at

*z* = − ∞. The undisturbed surface is at

*z* = 0, and the disturbed or wavy surface is at

*z* = η, where η is small in magnigude. If no fluid is to leak out of the bottom, we must have the condition

. Hence, Ψ =

*A**e**k**z* on

, where

*A* and the wave speed

*c* are constants to be determined from conditions at the interface.

*The free-surface condition:* At the free surface

, the kinematic condition holds:

Linearizing, this is simply

where the velocity

is linearized on to the surface

. Using the normal-mode and streamfunction representations, this condition is

, the second interfacial condition.

*Pressure relation across the interface:* For the case with

surface tension, the pressure difference over the interface at

*z* = η is given by the

Young–Laplace equation:

where

*σ* is the surface tension and

*κ* is the

curvature of the interface, which in a linear approximation is

Thus,

However, this condition refers to the total pressure (base+perturbed), thus

(As usual, The perturbed quantities can be linearized onto the surface

*z=0*.) Using

hydrostatic balance, in the form

*P* = − ρ

*g**z* + Const.,

this becomes

The perturbed pressures are evaluated in terms of streamfunctions, using the horizontal momentum equation of the linearised

Euler equations for the perturbations,

to yield

*p*' = ρ

*c**D*Ψ.

Putting this last equation and the jump condition together,

Substituting the second interfacial condition

and using the normal-mode representation, this relation becomes

*c*2ρ

*D*Ψ =

*g*Ψρ + σ

*k*2Ψ.

Using the solution Ψ =

*e**k**z*, this gives

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