Steep gravity waves in water of arbitrary uniform depth

1977/07/18 by Edward D. Cokelet · 6 citations
Earth and Planetary Sciences · Mathematics · #Ocean Waves and Remote Sensing #Coastal and Marine Dynamics #Oceanographic and Atmospheric Processes #Wavelength #Crest #Amplitude #Waves and shallow water #Swell #Curvature #Momentum (technical analysis) #Gravitational wave #Deep water #Wind wave #Gravity wave #Perturbation (astronomy) #Mechanical wave #Mechanics #Geometry #Physics #Wave propagation #Optics #Mathematics #Geology #Longitudinal wave #Quantum mechanics #Thermodynamics

paper · doi:10.1098/rsta.1977.0113

openalex publication_date 1977/07/18 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/18

Abstract

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.

Citations

• The singularity at the crest of a finite amplitude progressive Stokes wave
• Computer extension and analytic continuation of Stokes’ expansion for gravity waves
• The deformation of steep surface waves on water - I. A numerical method of computation
• D�termination rigoureuse des ondes permanentes d'ampleur finie
• Contributions to the theory of stokes waves
• The disintegration of wave trains on deep water Part 1. Theory
• On the mass, momentum, energy and circulation of a solitary wave. II
• Integral properties of periodic gravity waves of finite amplitude
• Détermination rigoureuse des ondes irrotationelles périodiques dans un canal à profondeur finie

Cited by

• The instabilities of gravity waves of finite amplitude in deep water I. Superharmonics
• Limiting gravity waves in water of finite depth
• Three-Dimensional Instability of Finite-Amplitude Water Waves
• On the existence of a wave of greatest height and Stokes’s conjecture
• Strongly Nonlinear Waves
• Stokes wave [wikipedia]