Editing Capillary wave (section)

====Derivation====
As [[Richard Feynman]] put it, "''[water waves] that are easily seen by everyone and which are usually used as an example of waves in elementary courses [...] are the worst possible example [...]; they have all the complications that waves can have.''"<ref>[[Richard Feynman|R.P. Feynman]], R.B. Leighton, and M. Sands (1963). ''[[The Feynman Lectures on Physics]].'' Addison-Wesley. Volume I, Chapter 51-4.</ref> The derivation of the general dispersion relation is therefore quite involved.<ref>See e.g. Safran (1994) for a more detailed description.</ref>

<!-- Therefore, first the assumptions involved are pointed out. <<  ? -->There are three contributions to the energy, due to gravity, to [[surface tension]], and to [[hydrodynamics]]. The first two are potential energies, and responsible for the two terms inside the parenthesis, as is clear from the appearance of <math>g</math> and <math>\sigma</math>. For gravity, an assumption is made of the density of the fluids being constant (i.e., incompressibility), and likewise <math>g</math> (waves are not high enough for gravitation to change appreciably). For surface tension, the deviations from planarity (as measured by derivatives of the surface) are supposed to be small. For common waves both approximations are good enough.

The third contribution involves the [[kinetic energy|kinetic energies]] of the fluids. It is the most complicated and calls for a [[hydrodynamics|hydrodynamic]] framework. Incompressibility is again involved (which is satisfied if the speed of the waves is much less than the speed of sound in the media), together with the flow being [[irrotational]] – the flow is then [[potential flow|potential]]. These are typically also good approximations for common situations.

The resulting equation for the potential (which is [[Laplace equation]]) can be solved with the proper boundary conditions. On one hand, the velocity must vanish well below the surface (in the "deep water" case, which is the one we consider, otherwise a more involved result is obtained, see [[Ocean surface wave#Science of waves|Ocean surface waves]].) On the other, its vertical component must match the motion of the surface. This contribution ends up being responsible for the extra  <math>k</math> outside the parenthesis, which causes '''all''' regimes to be dispersive, both at low values of <math>k</math>, and high ones (except around the one value at which the two dispersions cancel out.)

{| class="toccolours collapsible collapsed" width="90%" style="text-align:left"
!Dispersion relation for gravity–capillary waves on an interface between two semi–infinite fluid domains
|-
|Consider two fluid domains, separated by an interface with surface tension. The mean interface position is horizontal. It separates the upper from the lower fluid, both having a different constant mass density, <math>\rho</math> and <math>\rho'</math> for the lower and upper domain respectively. The fluid is assumed to be [[inviscid]] and [[Incompressible flow|incompressible]], and the flow is assumed to be [[irrotational]]. Then the flows are [[potential flow|potential]], and the velocity in the lower and upper layer can be obtained from <math>\nabla \phi</math> and <math>\nabla \phi'</math>, respectively. Here <math>\phi(x,y,z,t)</math> and <math>\phi'(x,y,z,t)</math> are [[potential flow|velocity potentials]].

Three contributions to the energy are involved: the [[potential energy]] <math>V_{g}</math> due to [[standard gravity|gravity]], the potential energy <math>V_{st}</math> due to the [[surface tension]] and the [[kinetic energy]] <math>T</math> of the flow. The part <math>V_{g}</math> due to gravity is the simplest: integrating the potential energy density due to gravity, <math>\rho g z</math> (or <math>\rho' g z</math>) from a reference height to the position of the surface, <math>z = \eta(x,y,t)</math>:<ref>Lamb (1994), §174 and §230.</ref>

:<math>
  V_\mathrm{g}
    = \iint dx\, dy\; \int_0^\eta dz\; (\rho - \rho') g z 
    = \frac{1}{2} (\rho-\rho') g \iint dx\, dy\; \eta^2,
</math>

assuming the mean interface position is at <math>z=0</math>.

An increase in area of the surface causes a proportional increase of energy due to surface tension:<ref name=LambCap>Lamb (1994), §266.</ref>

:<math>
  V_\mathrm{st}
  = \sigma \iint dx\, dy\;  
    \left[ 
      \sqrt{ 1 + \left( \frac{\partial \eta}{\partial x} \right)^2
               + \left( \frac{\partial \eta}{\partial y} \right)^2} 
      - 1
    \right]
  \approx \frac{1}{2} \sigma \iint dx\, dy\; 
    \left[ 
      \left( \frac{\partial \eta}{\partial x} \right)^2
      +
      \left( \frac{\partial \eta}{\partial y} \right)^2 
    \right],
</math>

where the first equality is the area in this ([[Gaspard Monge|Monge]]'s) representation, and the second
applies for small values of the derivatives (surfaces not too rough).

The last contribution involves the [[kinetic energy]] of the fluid:<ref name=LambKin>Lamb (1994), §61.</ref>

:<math>
  T= 
  \frac{1}{2} \iint dx\, dy\;  
  \left[ 
    \int_{-\infty}^\eta dz\; \rho\,  \left| \mathbf\nabla \Phi  \right|^2
    +
    \int_\eta^{+\infty} dz\; \rho'\, \left| \mathbf\nabla \Phi' \right|^2
  \right].
</math>

Use is made of the fluid being incompressible and its flow is irrotational (often, sensible approximations). As a result, both <math>\phi(x,y,z,t)</math> and <math>\phi'(x,y,z,t)</math> must satisfy the [[Laplace equation]]:<ref>Lamb (1994), §20</ref>

:<math>\nabla^2 \Phi = 0</math> &nbsp; and &nbsp; <math>\nabla^2 \Phi' = 0.</math>

These equations can be solved with the proper boundary conditions: <math>\phi</math> and <math>\phi'</math> must vanish well away from the surface (in the "deep water" case, which is the one we consider).

Using [[Green's identity]], and assuming the deviations of the surface elevation to be small (so the ''z''–integrations may be approximated by integrating up to <math>z=0</math> instead of <math>z = \eta</math>), the kinetic energy can be written as:<ref name=LambKin/>

:<math>
  T \approx
  \frac{1}{2} \iint dx\, dy\;  
  \left[
    \rho\,  \Phi\,  \frac{\partial \Phi }{\partial z}\; 
    -\; 
    \rho'\, \Phi'\, \frac{\partial \Phi'}{\partial z}
  \right]_{\text{at } z=0}.
</math>

To find the dispersion relation, it is sufficient to consider a [[sinusoidal]] wave on the interface, propagating in the ''x''–direction:<ref name=LambCap/>

:<math>\eta = a\, \cos\, ( kx - \omega t) = a\, \cos\, \theta ,</math>

with amplitude <math>a</math> and wave [[phase (waves)|phase]] <math>\theta = kx - \omega t</math>. The kinematic boundary condition at the interface, relating the potentials to the interface motion, is that the vertical velocity components must match the motion of the surface:<ref name=LambCap/>

:<math>\frac{\partial\Phi}{\partial z} = \frac{\partial\eta}{\partial t}</math>  &nbsp; and &nbsp; <math>\frac{\partial\Phi'}{\partial z} = \frac{\partial\eta}{\partial t}</math> &nbsp; at <math>z = 0</math>.

To tackle the problem of finding the potentials, one may try [[separation of variables]], when both fields can be expressed as:<ref name=LambCap/>

:<math>
\begin{align}
  \Phi(x,y,z,t) & = + \frac{1}{|k|} \text{e}^{+|k|z}\, 
                      \omega a\, \sin\, \theta,
  \\
  \Phi'(x,y,z,t)& = - \frac{1}{|k|} \text{e}^{-|k|z}\, 
                      \omega a\, \sin\, \theta.
\end{align}
</math>

Then the contributions to the wave energy, horizontally integrated over one wavelength <math>\lambda = 2\pi/k</math> in the ''x''–direction, and over a unit width in the ''y''–direction, become:<ref name=LambCap/><ref>Lamb (1994), §230.</ref>

:<math>
\begin{align}
  V_\text{g} &= \frac{1}{4} (\rho-\rho') g a^2 \lambda,
  \\
  V_\text{st} &= \frac{1}{4} \sigma k^2 a^2 \lambda,
  \\
  T &= \frac{1}{4} (\rho+\rho') \frac{\omega^2}{|k|} a^2 \lambda.
\end{align}
</math>

The dispersion relation can now be obtained from the [[Lagrangian mechanics|Lagrangian]] <math>L = T - V</math>, with <math>V</math> the sum of the potential energies by gravity <math>V_{g}</math> and surface tension <math>V_{st}</math>:<ref name=Whitham>{{cite book | first=G. B. | last=Whitham | author-link=Gerald B. Whitham | title=Linear and nonlinear waves | publisher = Wiley-Interscience | year=1974 | isbn=0-471-94090-9 }} See section 11.7.</ref>

:<math>
  L = \frac{1}{4} \left[
      (\rho+\rho') \frac{\omega^2}{|k|} - (\rho-\rho') g - \sigma k^2
    \right] a^2 \lambda.
</math>

For sinusoidal waves and linear wave theory, the [[averaged Lagrangian|phase–averaged Lagrangian]] is always of the form <math>L = D(\omega, k) a^{2}</math>, so that variation with respect to the only free parameter, <math>a</math>, gives the dispersion relation  <math>D(\omega, k) = 0</math>.<ref name=Whitham/> In our case <math>D(\omega,k)</math> is just the expression in the square brackets, so that the dispersion relation is:
:<math>
  \omega^2 = |k| \left( \frac{\rho-\rho'}{\rho+\rho'}\, g  + \frac{\sigma}{\rho+\rho'}\, k^2 \right),
</math>

the same as above.

As a result, the average wave energy per unit horizontal area, <math>(T + V)/\lambda</math>, is:

:<math>
  \bar{E} = \frac{1}{2}\, \left[ (\rho-\rho')\, g + \sigma k^2 \right]\, a^2.
</math>

As usual for linear wave motions, the potential and kinetic energy are equal (''[[equipartition theorem|equipartition]]'' holds): <math>T = V</math>.<ref>{{cite journal | title=On progressive waves | author=Lord Rayleigh (J. W. Strutt) | author-link=Lord Rayleigh | year=1877 | journal=Proceedings of the London Mathematical Society | volume=9 | pages=21–26 | doi=10.1112/plms/s1-9.1.21 | url=https://zenodo.org/record/1447762 }} Reprinted as Appendix in: ''Theory of Sound'' '''1''', MacMillan, 2nd revised edition, 1894.</ref>
|}