Deep-water capillary-gravity dispersion relation ($c^2=\frac{g}{k}+\frac{T}{\rho }k$)

A harmonic velocity potential decaying with depth, together with linearized kinematic and Bernoulli boundary conditions and capillary pressure, gives ${\omega }^2=gk+(T/\rho )k^3$. The corresponding phase speed combines long-wave gravity and short-wave capillarity contributions.