Capillary wave

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

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

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:

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

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

Gravity wave regime

For large wavelengths (small k = 2\pi/\lambda), 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 k) 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 \lambda_{m} are dominated by surface tension, and much above by gravity. The value of this wavelength and the associated minimum phase speed c_{m} are:

: \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'} }. For the air–water interface, \lambda_{m} is found to be 1.7 cm, and c_{m} is 0.23 m/s.

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.

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." The derivation of the general dispersion relation is therefore quite involved.

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 \sigma. For gravity, an assumption is made of the density of the fluids being constant (i.e., incompressibility), and likewise g (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 energies of the fluids. It is the most complicated and calls for a 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. 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 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.)

Gallery

File:Surface waves and water striders.JPG|Ripples on water created by water striders File:Narvijärvi ripples.jpg|Light breeze ripples in the surface water of a lake File:Black alder twig and water ripples at Wolfsberger Mühle, 2019-06-29.jpg|Ripples on water surface caused by a gas bubble

Notes

References

References

1. Lamb (1994), §267, page 458–460.
2. Dingemans (1997), Section 2.1.1, p. 45.
   Phillips (1977), Section 3.2, p. 37.
3. Falkovich, G.. (2011). "Fluid Mechanics, a short course for physicists". Cambridge University Press.
4. [[Richard Feynman. R.P. Feynman]], R.B. Leighton, and M. Sands (1963). ''[[The Feynman Lectures on Physics]].'' Addison-Wesley. Volume I, Chapter 51-4.
5. See e.g. Safran (1994) for a more detailed description.
6. Lamb (1994), §174 and §230.
7. Lamb (1994), §266.
8. Lamb (1994), §61.
9. Lamb (1994), §20
10. Lamb (1994), §230.
11. Whitham, G. B.. (1974). "Linear and nonlinear waves". Wiley-Interscience.
12. Lord Rayleigh (J. W. Strutt). (1877). "On progressive waves". Proceedings of the London Mathematical Society.