Steep gravity waves in water of arbitrary uniform depth

TLDR

Modern applications of water‑wave studies and recent theoretical developments highlight the need for a systematic, accurate calculation of the characteristics of steady, progressive gravity waves of finite amplitude in water of arbitrary uniform depth. The paper calculates the speed, momentum, energy and other integral properties of these waves using series expansions in a perturbation parameter that spans the full range of wave amplitudes. The authors extend the series to high order, sum them with Padé approximants, and tabulate the resulting speed, energy and other integral properties across all wave steepnesses and depth‑to‑wavelength ratios. The study finds that the highest wave is not the fastest, that energy, momentum and fluxes peak for waves lower than the highest, confirming earlier solitary and deep‑water results, and that the almost‑steepest deep‑water wave profile dips below a slightly less steep wave over most of its wavelength, with integration confirming an intermediate energy maximum.

Abstract

Modern applications of water-wave studies, as well as some recent theoretical developments, have shown the need for a systematic and accurate calculation of the characteristics of steady, progressive gravity waves of finite amplitude in water of arbitrary uniform depth. In this paper the speed, momentum, energy and other integral properties are calculated accurately by means of series expansions in terms of a perturbation parameter whose range is known precisely and encompasses waves from the lowest to the highest possible. The series are extended to high order and summed with Padé approximants. For any given wavelength and depth it is found that the highest wave is not the fastest. Moreover the energy, momentum and their fluxes are found to be greatest for waves lower than the highest. This confirms and extends the results found previously for solitary and deep-water waves. By calculating the profile of deep-water waves we show that the profile of the almost-steepest wave, which has a sharp curvature at the crest, intersects that of a slightly less-steep wave near the crest and hence is lower over most of the wavelength. An integration along the wave profile cross-checks the Padé-approximant results and confirms the intermediate energy maximum. Values of the speed, energy and other integral properties are tabulated in the appendix for the complete range of wave steepnesses and for various ratios of depth to wavelength, from deep to very shallow water.