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.

In the specified thin-sheet deep-water model, bending stiffness adds restoring force while sheet mass adds inertia. Their ratio to an uncovered gravity-capillary wave gives the displayed expression. Added inertia first lowers frequency at small wavenumber; at formally large wavenumber bending produces a frequency ratio proportional to $k^{1/2}$. That formal asymptote must not be extrapolated past the range where a thin-sheet continuum description is valid.

Write $\beta =D/{\rho }_w$ and $\alpha =m/{\rho }_w$ in the dispersion relation of a floating elastic plate. A stationary point of squared phase velocity satisfies $\beta k^4(3+2\alpha k)=g(1+2\alpha k)$. If plate inertia is neglected, its solution is $k=(g/(3\beta ){)}^{1/4}$. A steadily translating load can resonate only when its speed matches a phase velocity; this threshold differs from the group-velocity minimum of a flexural-gravity wave.

For bending stiffness $D$ and mass per area $m$, a plane wave of surface displacement $\eta$ satisfies $m{\eta }_{tt}+D{\nabla }_H^4\eta =p$. An inviscid deep-water velocity potential has vertical dependence $e^{kz}$; the kinematic boundary condition and linear Bernoulli equation give $p=({\rho }_w{\omega }^2/k-{\rho }_{w}g)\eta$. Eliminating $p$ gives the displayed dispersion relation. With water depth $d$, replace it by ${\omega }^2=(gk+D{k}^5/{\rho }_w)\tanh (kd)/[1+(m/{\rho }_w)k\tanh (kd)]$.

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.