Capillary wave
A capillary wave is a
Capillary waves are common in
A longer wavelength on a fluid interface will result in gravity–capillary waves which are influenced by both the effects of surface tension and gravity, as well as by fluid inertia. Ordinary gravity waves have a still longer wavelength.
When generated by light wind in open water, a nautical name for them is cat's paw waves. Light breezes which stir up such small ripples are also sometimes referred to as cat's paws. On the open ocean, much larger ocean surface waves (seas and swells) 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
where is the angular frequency, the surface tension, the density of the heavier fluid, the density of the lighter fluid and the wavenumber. The wavelength is For the boundary between fluid and vacuum (free surface), the dispersion relation reduces to
Gravity–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:[1][2]
where is the acceleration due to gravity, and are the
Gravity wave regime
For large wavelengths (small ), only the first term is relevant and one has gravity waves. 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 ) waves (e.g. 2 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 are dominated by surface tension, and much above by gravity. The value of this wavelength and the associated minimum phase speed are:[1]
For the
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 which corresponds to the minimal group velocity.[3]
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."[4] The derivation of the general dispersion relation is therefore quite involved.[5]
There are three contributions to the energy, due to gravity, to
The third contribution involves the
The resulting equation for the potential (which is
Dispersion relation for gravity–capillary waves on an interface between two semi–infinite fluid domains |
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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, and for the lower and upper domain respectively. The fluid is assumed to be irrotational. Then the flows are potential , and the velocity in the lower and upper layer can be obtained from and , respectively. Here and are velocity potentials.
Three contributions to the energy are involved: the potential energy due to gravity, the potential energy due to the surface tension and the kinetic energy of the flow. The part due to gravity is the simplest: integrating the potential energy density due to gravity, (or ) from a reference height to the position of the surface, :[6] assuming the mean interface position is at . An increase in area of the surface causes a proportional increase of energy due to surface tension:[7] where the first equality is the area in this (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:[8] Use is made of the fluid being incompressible and its flow is irrotational (often, sensible approximations). As a result, both and must satisfy the Laplace equation:[9]
These equations can be solved with the proper boundary conditions: and 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 instead of ), the kinetic energy can be written as:[8]
To find the dispersion relation, it is sufficient to consider a sinusoidal wave on the interface, propagating in the x–direction:[7]
with amplitude and wave phase . 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:[7]
To tackle the problem of finding the potentials, one may try separation of variables, when both fields can be expressed as:[7] Then the contributions to the wave energy, horizontally integrated over one wavelength in the x–direction, and over a unit width in the y–direction, become:[7][10] The dispersion relation can now be obtained from the Lagrangian , with the sum of the potential energies by gravity and surface tension :[11] For sinusoidal waves and linear wave theory, the phase–averaged Lagrangian is always of the form , so that variation with respect to the only free parameter, , gives the dispersion relation .[11] In our case is just the expression in the square brackets, so that the dispersion relation is: the same as above. As a result, the average wave energy per unit horizontal area, , is: As usual for linear wave motions, the potential and kinetic energy are equal (equipartition holds): .[12] |
See also
- Capillary action
- Dispersion (water waves)
- Fluid pipe
- Ocean surface wave
- Thermal capillary wave
- Two-phase flow
- Wave-formed ripple
Gallery
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Ripples on water created bywater striders
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Light breeze ripples in the surface water of a lake
Notes
- ^ a b c Lamb (1994), §267, page 458–460.
- ^ Dingemans (1997), Section 2.1.1, p. 45.
Phillips (1977), Section 3.2, p. 37. - ISBN 978-1-107-00575-4.
- ^ R.P. Feynman, R.B. Leighton, and M. Sands (1963). The Feynman Lectures on Physics. Addison-Wesley. Volume I, Chapter 51-4.
- ^ See e.g. Safran (1994) for a more detailed description.
- ^ Lamb (1994), §174 and §230.
- ^ a b c d e Lamb (1994), §266.
- ^ a b Lamb (1994), §61.
- ^ Lamb (1994), §20
- ^ Lamb (1994), §230.
- ^ ISBN 0-471-94090-9. See section 11.7.
- . Reprinted as Appendix in: Theory of Sound 1, MacMillan, 2nd revised edition, 1894.
References
- S2CID 119740891.
- ISBN 978-0-521-45868-9.
- ISBN 0-521-29801-6.
- Dingemans, M. W. (1997). Water wave propagation over uneven bottoms. Advanced Series on Ocean Engineering. Vol. 13. World Scientific, Singapore. pp. 2 Parts, 967 pages. ISBN 981-02-0427-2.
- Safran, Samuel (1994). Statistical thermodynamics of surfaces, interfaces, and membranes. Addison-Wesley.
- Tufillaro, N. B.; Ramshankar, R.; Gollub, J. P. (1989). "Order-disorder transition in capillary ripples". Physical Review Letters. 62 (4): 422–425. PMID 10040229.