Editing Capillary wave (section)

====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 <math>\lambda_{m}</math> are dominated by surface tension, and much above by gravity. The value of this wavelength and the associated minimum phase speed <math>c_{m}</math> are:<ref name=Lamb/>

:<math>
  \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'} }.
</math>
For the [[air]]–[[water]] interface, <math>\lambda_{m}</math> is found to be {{convert|1.7|cm|in|abbr=on}}, and <math>c_{m}</math> is {{convert|0.23|m/s|ft/s|abbr=on}}.<ref name=Lamb/>

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 (optics)|caustic]] which corresponds to the minimal group velocity.<ref>{{cite book |last=Falkovich |first=G. |title=Fluid Mechanics, a short course for physicists |publisher=Cambridge University Press |year=2011 |isbn=978-1-107-00575-4 |no-pp=yes |pages=Section 3.1 and Exercise 3.3}}</ref>