= Minimum phase speed of a gravity-capillary wave
{title2=$c_{\min}=(4g\sigma/\rho)^{1/4}$}
For a deep-water <gravity-capillary wave>, $c_p^2=g/k+\sigma k/\rho$ for $k>0$. Differentiation gives the unique minimum at $k_*=\sqrt{\rho g/\sigma}$, or <wavelength> $2\pi\sqrt{\sigma/(\rho g)}$. At this point the gravitational and capillary contributions are equal and $c_{\min}=(4g\sigma/\rho)^{1/4}$. Since $c_g=c_p+k\,dc_p/dk$, the <group velocity> equals the <phase velocity> at this minimum, despite their usual difference in a dispersive wave.
Back to article page
