Minimum phase speed of a flexural-gravity wave ($c_{p,\min }={\min }_{k>0}\omega (k)/k$)

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.