Part I: Symmetry Breaking of Water Waves. Part II: On the Superharmonic Instability of Surface Water Waves

Abstract

PART I: A weakly nonlinear Hamiltonian model for two dimensional irrotational waves on water of finite depth is developed. The truncated model is used to study families of periodic travelling waves of permanent form. It is shown that nonsymmetric periodic waves exist, which appear via spontaneous symmetry breaking bifurcations from symmetric waves. In order to check these results with the full water wave equations, two different methods are used to calculate nonsymmetric gravity waves on deep water. It is found that they exist and the structure of the bifurcation tree is the same as the one found for waves on water of finite depth using the weakly nonlinear Hamiltonian model. One of the methods is based on the quadratic relations between the Stokes coefficients discovered by Longuet-Higgins (1978a). The other method is a new one based on the Hamiltonian structure of the water wave problem. Another weakly nonlinear model is developed from the Hamiltonian formulation of water waves to study the bifurcation structure of gravity-capillary waves on water of finite depth. It is found that, besides a very rich structure of symmetric solutions, nonsymmetric Wilton ripples exist. They appear via spontaneous symmetry breaking bifurcation from symmetric solutions. The bifurcation tree is similar to that for gravity waves. The solitary wave with surface tension is studied with the same model close to a critical depth. It is found that the solution is not unique, and further nonsymmetric solitary waves are possible. The bifurcation tree has the same structure as for the case of periodic waves. The possibility of checking these results in low gravity experiments is discussed. PART II: Saffman's (1985) theory of the superharmonic stability of two-dimensional irrotational waves on fluid of infinite depth has been generalized to solitary and periodic waves of permanent form on fluid of finite uniform depth. The frame of reference for the calculation of the Hamiltonian for periodic waves of finite depth is found to be the frame in which the mean horizontal velocity is zero. Also, a simple analytical model has been constructed to demonstrate Saffman's (1985) theory. The model shows the change of geometrical and algebraic multiplicity of the eigenvalues and eigenvectors of the stability equation at the critical height. It confirms the existence of Hamiltonian systems with limit points at which there is no change of stability.