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

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.