Flexural-gravity wave (${\omega }^2=(gk+\beta k^5)/(1+\alpha k)$)

A flexural-gravity wave bends a floating elastic plate while displacing the underlying water. For a uniform plate with areal mass $m$ and bending stiffness $D$ above inviscid deep water of mass density ${\rho }_w$, put $\alpha =m/{\rho }_w$ and $\beta =D/{\rho }_w$. Combining the plate equation with the water's kinematic boundary condition gives the displayed dispersion relation. Both gravitational acceleration and plate bending restore displacement; water and plate inertia resist acceleration. It assumes small amplitude, no prestress and negligible shear deformation of the plate.

The group velocity of a flexural-gravity wave diverges on the long-wave gravity branch and on the short-wave bending branch, and therefore has a positive minimum between them when the bending stiffness is positive. This minimum bounds the propagation speed of narrow wave packets within the ideal model. It does not by itself determine a wind-generation threshold: steady forcing requires matching phase velocity, while dissipation and coupling determine growth.