1978/04/19 by M. S. Longuet‐Higgins · 4 citations
Earth and Planetary Sciences · Mathematics · #Ocean Waves and Remote Sensing #Oceanographic and Atmospheric Processes #Coastal and Marine Dynamics #Physics #Conservative vector field #Amplitude #Stream function #Instability #Classical mechanics #Mechanics #Perturbation (astronomy) #Nonlinear system #Velocity potential #Cartesian coordinate system #Phase velocity #Gravity wave #Mathematical analysis #Wave propagation #Mathematics #Optics #Geometry #Compressibility #Boundary value problem #Quantum mechanics
paper · doi:10.1098/rspa.1978.0080
openalex publication_date 1978/04/19 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/25
Abstract In this paper we embark on a calculation of all the normal-mode perturbations of nonlinear, irrotational gravity waves as a function of the wave steepness. The method is to use as coordinates the stream-function and velocity potential in the steady, unperturbed wave (seen in a reference frame moving with the phase speed) together with the time t. The dependent quantities are the cartesian displacements and the perturbed stream function at the free surface. To begin we investigate the ‘superharmonics’, i.e. those perturbations having the same horizontal scale as the fundamental wave, or less. When the steepness of the fundamental is small, the normal modes take the form of travelling waves superposed on the basic nonlinear wave. As the steepness increases the frequency of each perturbation tends generally to be diminished. At a steepness ak ≈ 0.436 it appears that the two lowest modes coalesce and an instability arises. There is evidence that this critical steepness corresponds precisely with the steepness at which the phase velocity is a maximum, considered as a function of ak. The calculations are facilitated by the discovery of some new identities between the coefficients in Stokes’s expansion for waves of finite amplitude.