# Capillary wave

A

**capillary wave**is awave travelling along the interface between two fluids, whose dynamics are dominated by the effects ofsurface tension .Capillary waves are common innature and home, and are often referred to as**ripple**. Thewavelength of capillary waves is typically less than a few centimeters.A

**gravity–capillary wave**on a fluid interface is influenced by both the effects of surface tension and gravity, as well as by the fluidinertia .**Capillary waves, proper**The

dispersion relation for capillary waves is:$omega^2=frac\{sigma\}\{\; ho+\; ho\text{'}\},\; |k|^3,$where "ω" is the

angular frequency , "σ" thesurface tension , "ρ" thedensity of the heavier fluid, "ρ"' the density of the lighter fluid and "k" thewavenumber . Thewavelength is $lambda=frac\{2\; pi\}\{k\}.$**Gravity–capillary waves**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 depthcite book | first=H. | last=Lamb | authorlink=Horace Lamb | year=1994 | title=Hydrodynamics | publisher=Cambridge University Press | edition=6

^{th}edition| isbn=9780521458689 §267, page 458–460.] [*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 Section 2.1.1, p. 45.*] :

cite book| first=O.M. | last=Phillips | title=The dynamics of the upper ocean |publisher=Cambridge University Press | year=1977 | edition=2^{nd}edition | isbn=0-521-29801-6 Section 3.2, p. 37.:$omega^2=|k|left(\; frac\{\; ho-\; ho\text{'}\}\{\; ho+\; ho\text{'}\}g+frac\{sigma\}\{\; ho+\; ho\text{'}\}k^2\; ight),$

where "g" is the acceleration due to gravity, "ρ" and "ρ‘" are the

mass density of the two fluids ("ρ > ρ‘").**Gravity wave regime**For large wavelengths (small "k = 2π/λ"), only the first term is relevant and one has

gravity waves .In this limit, the waves have agroup velocity half thephase 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 "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.

**Phase velocity minimum**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

_{c}" are dominated by surface tension, and much above by gravity. The value of this wavelength is:::$lambda\_c\; =\; 2\; pi\; sqrt\{\; frac\{sigma\}\{(\; ho-\; ho\text{'})\; g.$

For the

air –water interface, "λ_{c}" is found to be 1.7 cm.**Derivation**As

Richard Feynman put it, "... [water waves] , which are easily seen by everyone and which are used as an example of waves in elementary courses... are the worst possible example... they have all the complications that waves can have". [*R.P. Feynman, R.B. Leighton, and M. Sands "The Feynman lectures on physics" Addison-Wesley 1963. Section 51-4.*] The derivation of the general dispersion relation is therefore quite involved (see e.g. Ref. [*Samuel Safran "Statistical thermodynamics of surfaces, interfaces, and membranes" Addison-Wesley 1994.*] for a more detailed description.)Therefore, first the assumptions involved are pointed out. There are three contributions to the energy, due to gravity, to

surface tension , and tohydrodynamics . 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.The last contribution involves the kinetic energies of the fluids, and is the most involved. One must use a 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

potential; again, these are typically good approximations for common situations. The resulting equation for the potential (which isLaplace 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 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.)ext{e}^{-|k|z}, omega a, sin, heta.end{align}

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 [

*Lamb (1994), §230.*] ::$egin\{align\}\; V\_\; ext\{g\}\; =\; frac\{1\}\{4\}\; (\; ho-\; ho\text{'})\; g\; a^2\; lambda,\; \backslash \; V\_\; ext\{st\}\; =\; frac\{1\}\{4\}\; sigma\; k^2\; a^2\; lambda,\; \backslash \; T\; =\; frac\{1\}\{4\}\; (\; ho+\; ho\text{'})\; frac\{omega^2\}$

**Gallery****References***N. B. Tufillaro, R. Ramshankar, and J. P. Gollub, "Order-disorder transition in capillary ripples", "Physical Review Letters"

**62**(4), 422 (1989).**External links*** [

*http://ww.sklogwiki.org/SklogWiki/index.php/Capillary_waves Capillary waves entry at sklogwiki*]**ee also***

capillary action

*dispersion (water waves)

*thermal capillary wave

*two-phase flow

*ocean surface wave

*Wikimedia Foundation.
2010.*

### Look at other dictionaries:

**capillary wave**— ▪ oceanography small, free, surface water wave with such a short wavelength that its restoring force is the water s surface tension, which causes the wave to have a rounded crest and a V shaped trough. The maximum wavelength of a capillary… … Universalium**capillary wave**— kapiliarinė banga statusas T sritis fizika atitikmenys: angl. capillary wave vok. Kapillarwelle, f rus. капиллярная волна, f pranc. onde capillaire, f … Fizikos terminų žodynas**Thermal capillary wave**— Thermal motion is able to produce capillary waves at the molecular scale. At this scale,gravity and hydrodynamics can be neglected, and only the surface tension contribution isrelevant.Capillary wave theory (CWT) is a classic account of how… … Wikipedia**Capillary action**— Capillary action, capillarity, capillary motion, or wicking is the ability of a substance to draw another substance into it. The standard reference is to a tube in plants but can be seen readily with porous paper. It occurs when the adhesive… … Wikipedia**wave**— waveless, adj. wavelessly, adv. wavingly, adv. wavelike, adj. /wayv/, n., v., waved, waving. n. 1. a disturbance on the surface of a liquid body, as the sea or a lake, in the form of a moving ridge or swell. 2. any surging or progressing movement … Universalium**Wave**— /wayv/, n. a member of the Waves. Also, WAVE. [1942; see WAVES] * * * I In oceanography, a ridge or swell on the surface of a body of water, normally having a forward motion distinct from the motions of the particles that compose it. Ocean waves… … Universalium**Wave**— A wave is a disturbance that propagates through space and time, usually with transference of energy. While a mechanical wave exists in a medium (which on deformation is capable of producing elastic restoring forces), waves of electromagnetic… … Wikipedia**Wave radar**— Ocean surface waves can be measured by several radar remote sensing techniques. Several instruments based on a variety of different concepts and techniques are available to the user and these are all often called wave radars. This article (see… … Wikipedia**Gravity wave**— In fluid dynamics, gravity waves are waves generated in a fluid medium or at the interface between two media (e.g. the atmosphere or ocean) which has the restoring force of gravity or buoyancy.When a fluid parcel is displaced on an interface or… … Wikipedia**List of wave topics**— This is a list of wave topics.0 ndash;9*21 cm lineA*Abbe prism *absorption spectrum *acoustics *Airy disc *Airy wave theory *Alfvén wave *Alpha waves *amphidromic point *amplitude *amplitude modulation *analog sound vs. digital sound *animal… … Wikipedia