Editing Gravity wave (section)

===Deep water===
The [[phase velocity]] <math>c</math> of a linear gravity wave with [[wavenumber]] <math>k</math> is given by the formula

<math>c=\sqrt{\frac{g}{k}},</math>

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

<math>c=\sqrt{\frac{g}{k}+\frac{\sigma k}{\rho}},</math>

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

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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, <math>(u'(x,z,t),w'(x,z,t)).</math> Because the fluid is assumed incompressible, this velocity field has the [[streamfunction]] representation

:<math>\textbf{u}'=(u'(x,z,t),w'(x,z,t))=(\psi_z,-\psi_x),\,</math>

where the subscripts indicate [[partial derivatives]]. In this derivation it suffices to work in two dimensions <math>\left(x,z\right)</math>, 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 <math>\nabla\times\textbf{u}'=0.\,</math> In the streamfunction representation, <math>\nabla^2\psi=0.\,</math> Next, because of the translational invariance of the system in the ''x''-direction, it is possible to make the [[ansatz]]

:<math>\psi\left(x,z,t\right)=e^{ik\left(x-ct\right)}\Psi\left(z\right),\,</math>

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

:<math>\left(D^2-k^2\right)\Psi=0,\,\,\,\ D=\frac{d}{dz}.</math>

We work in a sea of infinite depth, so the boundary condition is at <math>\scriptstyle z=-\infty.</math> The undisturbed surface is at <math>\scriptstyle z=0</math>, and the disturbed or wavy surface is at <math>\scriptstyle z=\eta,</math> where <math>\scriptstyle\eta</math> is small in magnitude. If no fluid is to leak out of the bottom, we must have the condition

:<math>u=D\Psi=0,\,\,\text{on}\,z=-\infty.</math>

Hence, <math>\scriptstyle\Psi=Ae^{k z}</math> on <math>\scriptstyle z\in\left(-\infty,\eta\right)</math>, 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 <math>\scriptstyle z=\eta\left(x,t\right)\,</math>, the kinematic condition holds:

:<math>\frac{\partial\eta}{\partial t}+u'\frac{\partial\eta}{\partial x}=w'\left(\eta\right).\,</math>

Linearizing, this is simply

:<math>\frac{\partial\eta}{\partial t}=w'\left(0\right),\,</math>

where the velocity <math>\scriptstyle w'\left(\eta\right)\,</math> is linearized on to the surface <math>\scriptstyle z=0.\,</math> Using the normal-mode and streamfunction representations, this condition is <math>\scriptstyle c \eta=\Psi\,</math>, the second interfacial condition.

''Pressure relation across the interface'': For the case with [[surface tension]], the pressure difference over the interface at <math>\scriptstyle z=\eta</math> is given by the [[Young–Laplace]] equation:

:<math>p\left(z=\eta\right)=-\sigma\kappa,\,</math>

where ''σ'' is the surface tension and ''κ'' is the [[curvature]] of the interface, which in a linear approximation is

:<math>\kappa=\nabla^2\eta=\eta_{xx}.\,</math>

Thus,

:<math>p\left(z=\eta\right)=-\sigma\eta_{xx}.\,</math>

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

:<math>\left[P\left(\eta\right)+p'\left(0\right)\right]=-\sigma\eta_{xx}.</math>

(As usual, The perturbed quantities can be linearized onto the surface ''z=0''.) Using [[hydrostatic balance]], in the form <math>\scriptstyle P=-\rho g z+\text{Const.},</math>

this becomes

:<math>p=g\eta\rho-\sigma\eta_{xx},\qquad\text{on }z=0.\,</math>

The perturbed pressures are evaluated in terms of streamfunctions, using the horizontal momentum equation of the linearised [[Euler equations]] for the perturbations,

:<math>\frac{\partial u'}{\partial t} = - \frac{1}{\rho}\frac{\partial p'}{\partial x}\,</math>

to yield <math>\scriptstyle p'=\rho c D\Psi.</math>

Putting this last equation and the jump condition together,

:<math>c\rho D\Psi=g\eta\rho-\sigma\eta_{xx}.\,</math>

Substituting the second interfacial condition <math>\scriptstyle c\eta=\Psi\,</math> and using the normal-mode representation, this relation becomes <math>\scriptstyle c^2\rho D\Psi=g\Psi\rho+\sigma k^2\Psi.</math>

Using the solution <math>\scriptstyle \Psi=e^{k z}</math>, this gives

<math>c=\sqrt{\frac{g}{k}+\frac{\sigma k}{\rho}}.</math>

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Since <math>\scriptstyle c=\omega/k</math> is the phase speed in terms of the angular frequency <math>\omega</math> and the wavenumber, the gravity wave angular frequency can be expressed as

<math>\omega=\sqrt{gk}.</math>

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

<math>c_g=\frac{d\omega}{dk},</math>

and thus for a gravity wave,

<math>c_g=\frac{1}{2}\sqrt{\frac{g}{k}}=\frac{1}{2}c.</math>

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