Editing Capillary wave

{{short description|Wave on the surface of a fluid, dominated by surface tension}}
{{redirect|Rippled waves|the insect|Rippled wave|the electrical phenomenon|Ripple waveform||Ripple (disambiguation)}}
[[File:2006-01-14 Surface waves.jpg|thumb|Capillary waves (ripples) in water]]
[[File:Ripples Lifjord.jpg|thumb|Ripples on Lifjord in [[Øksnes Municipality]], [[Norway]]]]
[[File:Multy droplets impact.JPG|thumb|Capillary waves produced by [[droplet]] impacts on the interface between water and air.]]

A '''capillary wave''' is a [[wave]] traveling along the [[phase boundary]] of a fluid, whose [[Dynamics (mechanics)|dynamics]] and [[phase velocity]] are dominated by the effects of [[surface tension]].

Capillary waves are common in [[nature]], and are often referred to as '''ripples'''.  The [[wavelength]] of capillary waves on water is typically less than a few centimeters, with a [[phase speed]] in excess of 0.2–0.3 meter/second.

A longer wavelength on a fluid interface will result in '''gravity–capillary waves''' which are influenced by both the effects of surface tension and [[standard gravity|gravity]], as well as by fluid [[inertia]]. Ordinary [[gravity wave]]s have a still longer wavelength.

Light breezes upon the surface of water which stir up such small ripples are also sometimes referred to as 'cat's paws'. On the open ocean, much larger [[wind wave|ocean surface wave]]s ([[Wind wave|seas]] and [[swell (ocean)|swell]]s) may result from coalescence of smaller wind-caused ripple-waves.

==Dispersion relation==
The [[dispersion relation]] describes the relationship between [[wavelength]] and [[frequency]] in waves. Distinction can be made between pure capillary waves – fully dominated by the effects of surface tension – and gravity–capillary waves which are also affected by gravity.

===Capillary waves, proper===
The dispersion relation for capillary waves is

:<math>
\omega^2=\frac{\sigma}{\rho+\rho'}\, |k|^3,</math>
where <math>\omega</math> is the [[angular frequency]], <math>\sigma</math> the [[surface tension]], <math>\rho</math> the [[density]] of the 
heavier fluid, <math>\rho'</math> the density of the lighter fluid and <math>k</math> the [[wavenumber]]. The [[wavelength]] is 
<math>
\lambda=\frac{2 \pi}{k}.</math>
For the boundary between fluid and vacuum (free surface), the dispersion relation reduces to
:<math>
\omega^2=\frac{\sigma}{\rho}\, |k|^3.</math>

===Gravity–capillary waves===
[[File:Dispersion capillary.svg|thumb|right|Dispersion of gravity–capillary waves on the surface of deep water (zero mass density of upper layer, <math>\rho'=0</math>). Phase and group velocity divided by <math>\scriptstyle \sqrt[4]{g\sigma/\rho}</math> as a function of inverse relative wavelength <math>\scriptstyle \frac{1}{\lambda}\sqrt{\sigma/(\rho g)}</math>.<br>{{•}} Blue lines (A): phase velocity, Red lines (B): group velocity.<br>{{•}} Drawn lines: dispersion relation for gravity–capillary waves.<br>{{•}} Dashed lines: dispersion relation for deep-water gravity waves.<br>{{•}} Dash-dotted lines: dispersion relation valid for deep-water capillary waves.]]

When capillary waves are also affected substantially by gravity, they are called gravity–capillary waves. Their dispersion relation reads, for waves on the interface between two fluids of infinite depth:<ref name=Lamb>Lamb (1994), §267, page 458–460.</ref><ref>Dingemans (1997), Section 2.1.1, p.&nbsp;45.<br>Phillips (1977), Section 3.2, p.&nbsp;37.</ref>

:<math>
\omega^2=|k|\left( \frac{\rho-\rho'}{\rho+\rho'}g+\frac{\sigma}{\rho+\rho'}k^2\right),
</math>

where <math>g</math> is the acceleration due to [[standard gravity|gravity]], <math>\rho</math> and <math>\rho'</math> are the densities of the two fluids <math>(\rho > \rho')</math>. The factor <math>(\rho-\rho')/(\rho+\rho')</math> in the first term is the [[Atwood number]].

====Gravity wave regime====
{{further information|Airy wave theory}}

For large wavelengths (small <math>k = 2\pi/\lambda</math>), only the first term is relevant and one has [[gravity wave]]s.
In this limit, the waves have a [[group velocity]] half the [[phase velocity]]: following a single wave's crest in a group one can see the wave appearing at the back of the group, growing and finally disappearing at the front of the group.

====Capillary wave regime====
Shorter (large <math>k</math>) waves (e.g. 2&nbsp;mm for the water–air interface), which are proper capillary waves, do the opposite: an individual wave appears at the front of the group, grows when moving towards the group center and finally disappears at the back of the group. Phase velocity is two thirds of group velocity in this limit.

====Phase velocity minimum====
Between these two limits is a point at which the dispersion caused by gravity cancels out the dispersion due to the capillary effect.  At a certain wavelength, the group velocity equals the phase velocity, and there is no dispersion. At precisely this same wavelength, the phase velocity of gravity–capillary waves as a function of wavelength (or wave number) has a minimum. Waves with wavelengths much smaller than this critical wavelength <math>\lambda_{m}</math> are dominated by surface tension, and much above by gravity. The value of this wavelength and the associated minimum phase speed <math>c_{m}</math> are:<ref name=Lamb/>

:<math>
  \lambda_m = 2 \pi \sqrt{ \frac{\sigma}{(\rho-\rho') g}}
  \quad \text{and} \quad
  c_m = \sqrt{ \frac{2 \sqrt{ (\rho - \rho') g \sigma }}{\rho+\rho'} }.
</math>
For the [[air]]–[[water]] interface, <math>\lambda_{m}</math> is found to be {{convert|1.7|cm|in|abbr=on}}, and <math>c_{m}</math> is {{convert|0.23|m/s|ft/s|abbr=on}}.<ref name=Lamb/>

If one drops a small stone or droplet into liquid, the waves then propagate outside an expanding circle of fluid at rest; this circle is a [[caustic (optics)|caustic]] which corresponds to the minimal group velocity.<ref>{{cite book |last=Falkovich |first=G. |title=Fluid Mechanics, a short course for physicists |publisher=Cambridge University Press |year=2011 |isbn=978-1-107-00575-4 |no-pp=yes |pages=Section 3.1 and Exercise 3.3}}</ref>

====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>
|}

==See also==
* [[Capillary action]]
* [[Dispersion (water waves)]]
* [[Ocean surface wave]]
* [[Thermal capillary wave]]
* [[Two-phase flow]]
* [[Wave-formed ripple]]

==Gallery==
<gallery>
File:Surface waves and water striders.JPG|Ripples on water created by [[water strider]]s
File:Narvijärvi ripples.jpg|Light breeze ripples in the surface water of a lake
</gallery>

==Notes==
{{reflist|30em}}

==References==
*{{cite journal | last=Longuet-Higgins | first=M. S. | author-link=Michael S. Longuet-Higgins | title=The generation of capillary waves by steep gravity waves | journal=Journal of Fluid Mechanics | volume=16 | issue=1 | year=1963 | issn=1469-7645 | pages= 138–159 | doi=10.1017/S0022112063000641 |bibcode = 1963JFM....16..138L | s2cid=119740891 }}
*{{cite book | first=H. | last=Lamb | author-link=Horace Lamb | year=1994 | title=Hydrodynamics | publisher=Cambridge University Press | edition=6th| isbn=978-0-521-45868-9 }}
*{{cite book| first=O. M. | last=Phillips | author-link=Owen Martin Phillips |title=The dynamics of the upper ocean |publisher=Cambridge University Press | year=1977 | edition=2nd | isbn=0-521-29801-6 }}
*{{cite book | title=Water wave propagation over uneven bottoms | first=M. W. |last=Dingemans | year=1997 | series=Advanced Series on Ocean Engineering | volume=13 | publisher=World Scientific, Singapore | pages=2 Parts, 967 pages | isbn=981-02-0427-2 }}
*{{cite book | first=Samuel | last=Safran | title=Statistical thermodynamics of surfaces, interfaces, and membranes | publisher=Addison-Wesley | year=1994 }}
*{{Cite journal | first1=N. B. | last1=Tufillaro | first2=R. | last2=Ramshankar | first3=J. P. | last3=Gollub | title=Order-disorder transition in capillary ripples | journal=Physical Review Letters | volume=62 | issue=4 | pages=422–425 | year=1989 | doi=10.1103/PhysRevLett.62.422 | pmid=10040229 | bibcode=1989PhRvL..62..422T| url=http://scholarship.haverford.edu/cgi/viewcontent.cgi?article=1062&context=physics_facpubs | url-access=subscription }}

==External links==
{{commons category|Ripples (water waves)}}
*[http://www.sklogwiki.org/SklogWiki/index.php/Capillary_waves Capillary waves entry at sklogwiki]

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[[Category:Water waves]]
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