Internal resonant interactions of three free surface-waves in a circular cylindrical basin

Abstract

The basic equations of free capillary-gravity surface-waves in a circular cylindrical basin were derived from Luke's principle. Taking Galerkin's expansion of the velocity potential and the free surface elevation, the second-order perturbation equations were derived by use of expansion of multiple scale. The nonlinear interactions with the second order internal resonance of three free surface-waves were discussed based on the above. The results include: derivation of the couple equations of resonant interactions among three waves and the conservation laws; analysis of the positions of equilibrium points in phase plane; study of the resonant parameters and the non-resonant parameters respectively in all kinds of circumstances; derivation of the stationary solutions of the second-order interaction equations corresponding to different parameters and analysis of the stability property of the solutions; discussion of the effective solutions only in the limited time range. The analysis makes it clear that the energy transformation mode among three waves differs because of the different initial conditions under nontrivial circumstance. The energy may either exchange among three waves periodically or damp or increase in single waves.