Mass, momentum, and energy flux conservation between linear and nonlinear steady-state wave groups

Abstract

This paper provides a mass, momentum, and energy flux conservation analysis between the linear and nonlinear steady-state wave groups. Convergent high-order solutions for nonlinear wave groups with multiple steady-state near resonances in deep water have been obtained by means of the homotopy analysis method. The small divisors associated with nearly resonant components are transformed to singularities that are originally caused by exact resonances by a piecewise auxiliary linear operator. Both two primary components and other nearly resonant ones are considered in the initial guess to search for finite amplitude wave groups. It is found that as nonlinearity of wave groups increases, more wave components appear in the spectrum due to the nearly resonant interactions. The nonlinear wave fields change from the initial bi-chromatic waves that contain only two nontrivial primary components into the steady-state resonant waves that contain both two primary components and other nearly resonant ones. The conservation of mean rates of mass, momentum, and energy fluxes is established between the nonlinear wave groups and linear waves that are combined by two primary components with the same frequencies as in nonlinear wave groups. Comparison of the linear and nonlinear wave fields shows that the nearly resonant components influence the wave field distribution significantly: the nonlinear free surfaces have more peaked crests, steeper troughs, and more flatten wave nodes, and the related velocities at the crests and troughs increase more rapidly with the nonlinearity. All of these findings are helpful to enrich and deepen our understanding about nonlinear wave groups.