Capillary wave

File:Ripples Lifjord.jpg
Ripples on Lifjord in Øksnes, Norway.
File:Multy droplets impact.JPG
Capillary waves produced by droplet impacts.

A capillary wave is a wave travelling along the interface between two fluids, whose dynamics are dominated by the effects of surface tension. Capillary waves are common in nature and home, and are often referred to as ripple. The wavelength 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 fluid inertia.

Capillary waves, proper

The dispersion relation for capillary waves is

${\displaystyle \omega ^{2}={\frac {\sigma }{\rho +\rho '}}\,|k|^{3},}$

where ω is the angular frequency, σ the surface tension, ρ the density of the heavier fluid, ρ' the density of the lighter fluid and k the wavenumber. The wavelength is ${\displaystyle \lambda ={\frac {2\pi }{k}}.}$

Gravity–capillary waves

File:Dispersion capillary.svg
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 ${\displaystyle \scriptstyle {\sqrt[{4}]{g\sigma /\rho }}}$ as a function of inverse relative wavelength ${\displaystyle \scriptstyle {\frac {1}{\lambda }}{\sqrt {\sigma /(\rho g)}}}$.
Blue lines (A): phase velocity, Red lines (B): group velocity.
Drawn lines: dispersion relation for gravity–capillary waves.
Dashed lines: dispersion relation for deep-water gravity waves.
Dash-dot lines: dispersion relation valid for deep–water 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 depth[1][2]:

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

where g is the acceleration due to gravity, 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 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), 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 λc are dominated by surface tension, and much above by gravity. The value of this wavelength is[1]:

${\displaystyle \lambda _{c}=2\pi {\sqrt {\frac {\sigma }{(\rho -\rho ')g}}}.}$

For the airwater interface, λc is found to be 1.7 cm[1].

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" [3] The derivation of the general dispersion relation is therefore quite involved (see e.g. Ref. [4] 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 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.

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 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.)

References

1. Lamb, H. (1994). Hydrodynamics (6th edition ed.). Cambridge University Press. ISBN 9780521458689. §267, page 458–460.
2. Dingemans, M. W. (1997). Water wave propagation over uneven bottoms. Advanced Series on Ocean Engineering. 13. World Scientific, Singapore. pp. 2 Parts, 967 pages. ISBN 981-02-0427-2. Section 2.1.1, p. 45.
Phillips, O.M. (1977). The dynamics of the upper ocean (2nd edition ed.). Cambridge University Press. ISBN 0-521-29801-6. Section 3.2, p. 37.
3. R.P. Feynman, R.B. Leighton, and M. Sands "The Feynman lectures on physics" Addison-Wesley 1963. Section 51-4.
4. Samuel Safran "Statistical thermodynamics of surfaces, interfaces, and membranes" Addison-Wesley 1994.
5. Lamb (1994), §174 and §230.
6. Lamb (1994), §266.
7. Lamb (1994), §61.
8. Lamb (1994), §20
9. Lamb (1994), §230.
10. Whitham, G.B. (1974). Linear and nonlinear waves. Wiley-Interscience. ISBN 0 471 94090 9. See section 11.7.
11. Lord Rayleigh (J. W. Strutt) (1877). "On progressive waves". Proceedings of the London Mathematical Society. 9: 21–26. Reprinted as Appendix in: Theory of Sound 1, MacMillan, 2nd revised edition, 1894.