Minimum phase speed of a gravity-capillary wave ($c_{\min }=(4g\sigma /\rho {)}^{1/4}$)

For a deep-water gravity-capillary wave, $c_p^2=g/k+\sigma k/\rho$ for $k>0$. Differentiation gives the unique minimum at $k_{\ast }=\sqrt{\rho g/\sigma }$, 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 $c_g=c_p+kd{c}_p/dk$, the group velocity equals the phase velocity at this minimum, despite their usual difference in a dispersive wave.