Gravity Wave - Quantitative Description

Quantitative Description

Further information: Airy wave theory and Stokes wave

The phase velocity of a linear gravity wave with wavenumber is given by the formula

where g is the acceleration due to gravity. When surface tension is important, this is modified to

where σ is the surface tension coefficient and ρ is the density.

Details of the phase-speed derivation

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 The undisturbed surface is at, and the disturbed or wavy surface is at where is small in magnitude. If no fluid is to leak out of the bottom, we must have the condition

Hence, 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 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

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

Putting this last equation and the jump condition together,

Substituting the second interfacial condition and using the normal-mode representation, this relation becomes

Using the solution, this gives

Since is the phase speed in terms of the angular frequency and the wavenumber, the gravity wave angular frequency can be expressed as

The group velocity of a wave (that is, the speed at which a wave packet travels) is given by

and thus for a gravity wave,

The group velocity is one half the phase velocity. A wave in which the group and phase velocities differ is called dispersive.