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| {{two other uses|ripples on fluid interfaces|ripples in electricity|ripple (electrical)||ripple (disambiguation)}}
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| [[Image:2006-01-14 Surface waves.jpg|thumb|right|Capillary wave (ripple) in water]]
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| [[Image:Ripples Lifjord.jpg|thumb|right|Ripples on Lifjord in [[Øksnes]], [[Norway]]]]
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| [[Image:Multy droplets impact.JPG|thumb|right|Capillary waves produced by [[droplet]] impacts on the interface between water and air. In this photograph the origin of the popular term "cat's [[paw]]" is evident]]
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| A '''capillary wave''' is a [[wave]] traveling along the [[phase boundary]] of a fluid, whose [[Dynamics (mechanics)|dynamics]] are dominated by the effects of [[surface tension]].
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| Capillary waves are common in [[nature]] and at home, and are often referred to as '''ripples'''. The [[wavelength]] of capillary waves in water is typically less than a few centimeters.
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| When generated by light wind in open water, a nautical name for them is "cat's paw" waves, since they may resemble paw prints. Light breezes 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 (seas and [[swell (ocean)|swell]]s) may result from coalescence of smaller wind-caused ripple-waves.
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| A '''gravity–capillary wave''' on a fluid interface is influenced by both the effects of surface tension and [[standard gravity|gravity]], as well as by fluid [[inertia]]. | |
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| ==Capillary waves, proper==
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| The [[dispersion relation]] for capillary waves is
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| :<math> | |
| \omega^2=\frac{\sigma}{\rho+\rho'}\, |k|^3,</math>
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| where ''ω'' is the [[angular frequency]], ''σ'' the [[surface tension]], ''ρ'' the [[density]] of the
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| heavier fluid, ''ρ''' the density of the lighter fluid and ''k'' the [[wavenumber]]. The [[wavelength]] is
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| <math>
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| \lambda=\frac{2 \pi}{k}.</math>
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| ==Gravity–capillary waves== | |
| [[Image:Dispersion capillary.svg|thumb|right|Dispersion of gravity–capillary waves on the surface of deep water (zero mass density of upper layer, ''ρ′ = 0''). 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-dot lines: dispersion relation valid for deep–water capillary waves.]]
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| In general, waves are also affected by gravity and are then 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. 45.<br>Phillips (1977), Section 3.2, p. 37.</ref>
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| :<math>
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| \omega^2=|k|\left( \frac{\rho-\rho'}{\rho+\rho'}g+\frac{\sigma}{\rho+\rho'}k^2\right),
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| </math>
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| where ''g'' is the acceleration due to [[standard gravity|gravity]], ''ρ'' and ''ρ‘'' are the [[mass density]] of the two fluids (''ρ > ρ‘''). Notice the factor <math>(\rho-\rho')/(\rho+\rho')</math> in the first term is the [[Atwood number]].
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| ===Gravity wave regime===
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| {{further|Airy wave theory}}
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| For large wavelengths (small ''k = 2π/λ''), only the first term is relevant and one has [[gravity wave]]s.
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| 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.
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| ===Capillary wave regime===
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| Shorter (large ''k'') waves (e.g. 2 mm), 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.
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| ===Phase velocity minimum===
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| Between these two limits, an interesting and common situation occurs when 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 ''λ<sub>c</sub>'' are dominated by surface tension, and much above by gravity. The value of this wavelength is:<ref name=Lamb/>
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| ::<math>\lambda_c = 2 \pi \sqrt{ \frac{\sigma}{(\rho-\rho') g}}.</math>
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| For the [[air]]–[[water]] interface, ''λ<sub>c</sub>'' is found to be 1.7 cm.<ref name=Lamb/>
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| 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 caustics 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 |nopp=yes |pages=Section 3.1 and Exercise 3.3}}</ref>
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| ===Derivation===
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| 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>
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| 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 ''g'' and ''σ''. For gravity, an assumption is made of the density of the fluids being constant (i.e., incompressibility), and likewise ''g'' (waves are not high for gravitation to change appreciably). For surface tension, the deviations from planarity (as measured by derivatives of the surface) are supposed to be small. Both approximations are excellent for common waves.
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| The last contribution involves the [[kinetic energy|kinetic energies]] of the fluids, and is the most involved. One must use a [[hydrodynamics|hydrodynamic]] framework to tackle this problem. 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
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| [[potential flow|potential]]; again, these are typically 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 ''k'' outside the parenthesis, which causes '''all''' regimes to be dispersive, both at low values of ''k'', and high ones (except around the one value at which the two dispersions cancel out.)
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| {| class="toccolours collapsible collapsed" width="90%" style="text-align:left"
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| !Dispersion relation for gravity–capillary waves on an interface between two semi–infinite fluid domains
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| |-
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| |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, ''ρ'' and ''ρ’'' for the lower and upper domain respectively. The fluid is assumed to be [[inviscid]] and [[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 '''''∇'''Φ'' and '''''∇'''Φ’'', respectively. Here ''Φ(x,y,z,t)'' and ''Φ’(x,y,z,t)'' are [[potential flow|velocity potentials]].
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| Three contributions to the energy are involved: the [[potential energy]] ''V<sub>g</sub>'' due to [[standard gravity|gravity]], the potential energy ''V<sub>st</sub>'' due to the [[surface tension]] and the [[kinetic energy]] ''T'' of the flow. The part ''V<sub>g</sub>'' due to gravity is the simplest: integrating the potential energy density due to gravity, ''ρ g z'' (or ''ρ’ g z'') from a reference height to the position of the surface, ''z = η(x,y,t)'':<ref>Lamb (1994), §174 and §230.</ref>
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| :<math>
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| V_\mathrm{g}
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| = \iint dx\, dy\; \int_0^\eta dz\; (\rho - \rho') g z
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| = \frac{1}{2} (\rho-\rho') g \iint dx\, dy\; \eta^2,
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| </math>
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| assuming the mean interface position is at ''z=0''.
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| An increase in area of the surface causes a proportional increase of energy due to surface tension:<ref name=LambCap>Lamb (1994), §266.</ref>
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| :<math>
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| V_\mathrm{st}
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| = \sigma \iint dx\, dy\;
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| \left[
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| \sqrt{ 1 + \left( \frac{\partial \eta}{\partial x} \right)^2
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| + \left( \frac{\partial \eta}{\partial y} \right)^2}
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| - 1
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| \right]
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| \approx \frac{1}{2} \sigma \iint dx\, dy\;
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| \left[
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| \left( \frac{\partial \eta}{\partial x} \right)^2
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| +
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| \left( \frac{\partial \eta}{\partial y} \right)^2
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| \right],
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| </math>
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| where the first equality is the area in this ([[Gaspard Monge|Monge]]'s) representation, and the second
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| applies for small values of the derivatives (surfaces not too rough).
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| The last contribution involves the [[kinetic energy]] of the fluid:<ref name=LambKin>Lamb (1994), §61.</ref>
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| :<math>
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| T=
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| \frac{1}{2} \iint dx\, dy\;
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| \left[
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| \int_{-\infty}^\eta dz\; \rho\, \left| \bold\nabla \Phi \right|^2
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| +
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| \int_\eta^{+\infty} dz\; \rho'\, \left| \bold\nabla \Phi' \right|^2
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| \right].
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| </math>
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| Use is made of the fluid being incompressible and its flow is irrotational (often, sensible approximations). As a result, both ''Φ(x,y,z,t)'' and ''Φ’(x,y,z,t)'' must satisfy the [[Laplace equation]]:<ref>Lamb (1994), §20</ref>
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| :<math>\nabla^2 \Phi = 0</math> and <math>\nabla^2 \Phi' = 0.</math>
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| These equations can be solved with the proper boundary conditions: ''Φ'' and ''Φ’'' must vanish well away from the surface (in the "deep water" case, which is the one we consider).
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| 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 ''z=0'' instead of ''z=η''), the kinetic energy can be written as:<ref name=LambKin/>
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| :<math>
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| T \approx
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| \frac{1}{2} \iint dx\, dy\;
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| \left[
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| \rho\, \Phi\, \frac{\partial \Phi }{\partial z}\;
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| -\;
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| \rho'\, \Phi'\, \frac{\partial \Phi'}{\partial z}
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| \right]_{\text{at } z=0}.
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| </math>
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| To find the dispersion relation, it is sufficient to consider a [[sinusoidal]] wave on the interface, propagating in the ''x''–direction:<ref name=LambCap/>
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| :<math>\eta = a\, \cos\, ( kx - \omega t) = a\, \cos\, \theta ,</math>
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| with amplitude ''a'' and wave [[phase (waves)|phase]] ''θ = kx - ωt''. 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/> | |
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| :<math>\frac{\partial\Phi}{\partial z} = \frac{\partial\eta}{\partial t}</math> and <math>\frac{\partial\Phi'}{\partial z} = \frac{\partial\eta}{\partial t}</math> at ''z = 0''.
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| To tackle the problem of finding the potentials, one may try [[separation of variables]], when both fields can be expressed as:<ref name=LambCap/>
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| :<math>
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| \begin{align}
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| \Phi(x,y,z,t) & = + \frac{1}{|k|} \text{e}^{+|k|z}\,
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| \omega a\, \sin\, \theta,
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| \\
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| \Phi'(x,y,z,t)& = - \frac{1}{|k|} \text{e}^{-|k|z}\,
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| \omega a\, \sin\, \theta.
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| \end{align}
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| </math>
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| Then the contributions to the wave energy, horizontally integrated over one wavelength ''λ = 2π/k'' in the ''x''–direction, and over a unit width in the ''y''–direction, become:<ref name=LambCap/><ref>Lamb (1994), §230.</ref>
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| :<math>
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| \begin{align}
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| V_\text{g} &= \frac{1}{4} (\rho-\rho') g a^2 \lambda,
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| \\
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| V_\text{st} &= \frac{1}{4} \sigma k^2 a^2 \lambda,
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| \\
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| T &= \frac{1}{4} (\rho+\rho') \frac{\omega^2}{|k|} a^2 \lambda.
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| \end{align}
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| </math>
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| The dispersion relation can now be obtained from the [[Lagrangian]] ''L = T - V'', with ''V'' the sum of the potential energies by gravity ''V<sub>g</sub>'' and surface tension ''V<sub>st</sub>'':<ref name=Whitham>{{cite book | first=G. B. | last=Whitham | authorlink=Gerald B. Whitham | title=Linear and nonlinear waves | publisher = Wiley-Interscience | year=1974 | isbn=0-471-94090-9 }} See section 11.7.</ref>
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| :<math>
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| L = \frac{1}{4} \left[
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| (\rho+\rho') \frac{\omega^2}{|k|} - (\rho-\rho') g - \sigma k^2
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| \right] a^2 \lambda.
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| </math>
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| For sinusoidal waves and linear wave theory, the phase–averaged Lagrangian is always of the form ''L = D(ω,k) a²'', so that variation with respect to the only free parameter, ''a'', gives the dispersion relation ''D(ω,k) = 0''.<ref name=Whitham/> In our case ''D(ω,k)'' is just the expression in the square brackets, so that the dispersion relation is:
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| :<math>
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| \omega^2 = |k| \left( \frac{\rho-\rho'}{\rho+\rho'}\, g + \frac{\sigma}{\rho+\rho'}\, k^2 \right),
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| </math>
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| the same as above.
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| As a result, the average wave energy per unit horizontal area, ''( T + V ) / λ'', is:
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| :<math>
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| \bar{E} = \frac{1}{2}\, \left[ (\rho-\rho')\, g + \sigma k^2 \right]\, a^2.
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| </math>
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| As usual for linear wave motions, the potential and kinetic energy are equal (''equipartition'' holds): ''T = V''.<ref>{{cite journal | title=On progressive waves | author=Lord Rayleigh (J. W. Strutt) | authorlink=Lord Rayleigh | year=1877 | journal=Proceedings of the London Mathematical Society | volume=9 | pages=21–26 | doi=10.1112/plms/s1-9.1.21 }} Reprinted as Appendix in: ''Theory of Sound'' '''1''', MacMillan, 2nd revised edition, 1894.</ref>
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| |}
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| ==See also==
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| * [[Capillary action]]
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| * [[Dispersion (water waves)]]
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| * [[Fluid pipe]]
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| * [[Thermal capillary wave]]
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| * [[Two-phase flow]]
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| * [[Ocean surface wave]]
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| * [[Wave-formed ripple]]
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| ==Gallery==
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| <gallery>
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| Image:2006-01-14 Surface waves.jpg|Ripples on water
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| Image:Surface waves and water striders.JPG|Ripples on water created by [[water strider]]s
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| Image:Plughole.JPG|Ripples of tapwater over a plughole
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| Image:Ripple - in rail.jpg|
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| </gallery>
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| | |
| ==Notes==
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| {{reflist|2}}
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| | |
| ==References==
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| *{{cite book | first=H. | last=Lamb | authorlink=Horace Lamb | year=1994 | title=Hydrodynamics | publisher=Cambridge University Press | edition=6th| isbn=978-0-521-45868-9 }}
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| *{{cite book| first=O. M. | last=Phillips | title=The dynamics of the upper ocean |publisher=Cambridge University Press | year=1977 | edition=2nd | isbn=0-521-29801-6 }}
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| *{{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 }}
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| *{{cite book | first=Samuel | last=Safran | title=Statistical thermodynamics of surfaces, interfaces, and membranes | publisher=Addison-Wesley | year=1994 }}
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| *{{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}}
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| ==External links==
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| *[http://www.sklogwiki.org/SklogWiki/index.php/Capillary_waves Capillary waves entry at sklogwiki]
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| {{DEFAULTSORT:Capillary Wave}}
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| [[Category:Fluid dynamics]]
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| [[Category:Water waves]]
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| [[ar:مويجة]]
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| [[de:Kapillarwelle]]
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| [[pl:Fale kapilarne]]
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