# Capillary Wave

Capillary wave is a tiny wavlet a few centimeters in wavelength, a few millimeters in height, and with a period of less than half a second. Capillary waves are the most common type of water waves, and apparent when water is calm and flat. These tiny waves would raise when gentle wind breezes through water surface, and decay as soon as the wind dies out. Because they are tiny and superficial on the water surface, their restoring force is surface tension rather than gravity. The restoring force for larger waves, inculding ripples and tides, is gravity. [2]

## Disperison relation

The thermally excited displacement <math>\xi (r,t)</math> of the free surface of a liquid from the equilibrium normal to the surface can be Fourier-decomposed into a complete set of surface modes as [3]

- <math>

\xi (r,t)= \xi_0 \sum_k </math>exp<math>(i k \cdot r+\Omega t)</math>

The complex wave frequency <math>\Omega</math> is related to <math>k</math> via disperison relation.

- <math>

f^2=\frac{\gamma}{\rho_U+\rho_L}\, |k|^3,</math>

where <math>f</math> is the capillary wave frequency; <math>\gamma</math>, the surface tension; <math>\rho_U</math>, the density of the upper phase; <math>\rho_L</math> the density of the lower phase; and <math>k</math>, the wavevector of the capillary wave.

## References

[1] http://www.flickr.com/photos/arenamontanus/1358639911/

[2] Paul R. Pinet (2009). *Invitation to oceanography* (5th Ed). Jones and Bartlett Publishers.

[3] Alexander George Volkov (2001). *Liquid interfaces in chemical, biological, and pharmaceutical applications *. CRC Press. ISBN: 9780824704575