Past exam of the fluids and waves course of the University of Oxford / 2023 / 3 / c

In the deep-water limit $kh\gg 1$, $\tanh (kh)\sim 1$. Put $\beta =B/\rho >0$. The squared phase speed of the flexural-gravity wave is thenIt diverges as $k\to 0^{+}$ and as $k\to \infty$ in this limiting model. Its unique stationary point satisfiesThe second derivative $2g/k^3+6\beta k$ is positive. Using $g/k_{\ast }=3\beta k_{\ast }^3$ gives the minimum phase speed of a flexural-gravity wave:This is the deep-water minimum: applying it to a finite depth requires $k_{\ast }h\gg 1$. Otherwise the $\tanh (kh)$ factor must be retained in the minimization. Within the deep-water approximation, phase speed and group velocity coincide at $k_{\ast }$, although the wave family remains dispersive.

Define the dimensionless parameterThe stipulated hierarchy $kh\ll \delta \ll 1$ implies both $kh\ll 1$ andThus the finite-depth correction uses $\tanh (kh)\sim kh$, while the restoring effect of bending stiffness is small compared with gravity. The dispersion relation becomesHence, to leading order, these surface gravity waves are nondispersive waves withFor clarity, the first corrections to the phase speed areThey exhibit the weak residual wave dispersion at finite wavenumber. The claimed nondispersive behavior is the asymptotic leading-order result, not exact equality for the complete relation.