1979/09/11 by John D. Fenton · 129 citations
Earth and Planetary Sciences · Physics and Astronomy · #Ocean Waves and Remote Sensing #Oceanographic and Atmospheric Processes #Nonlinear Waves and Solitons #Cnoidal wave #Stokes wave #Wavelength #Dimensionless quantity #Wave shoaling #Waves and shallow water #Airy wave theory #Physics #Mechanics #Wave height #Surface wave #Breaking wave #Wave propagation #Mechanical wave #Optics #Longitudinal wave #Thermodynamics
paper · doi:10.1017/s0022112079000975
published in Journal of Fluid Mechanics 94(1), 129-161 (Cambridge University Press)
openalex publication_date 1979/09/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/08
A method is outlined by which high-order solutions are obtained for steadily progressing shallow water waves. It is shown that a suitable expansion parameter for these cnoidal wave solutions is the dimensionless wave height divided by the parameter m of the cn functions: this explicitly shows the limitation of the theory to waves in relatively shallow water. The corresponding deep water limitation for Stokes waves is analysed and a modified expansion parameter suggested. Cnoidal wave solutions to fifth order are given so that a steady wave problem with known water depth, wave height and wave period or length may be solved to give expressions for the wave profile and fluid velocities, as well as integral quantities such as wave power and radiation stress. These series solutions seem to exhibit asymptotic behaviour such that there is no gain in including terms beyond fifth order. Results from the present theory are compared with exact numerical results and with experiment. It is concluded that the fifth-order cnoidal theory should be used in preference to fifth-order Stokes wave theory for wavelengths greater than eight times the water depth, when it gives quite accurate results.