A Low-Dimensional Model for the Maximal Amplification Factor of Bichromatic Wave Groups
We consider a low-dimensional model derived from the nonlinear-Schrödinger equation that describes the evolution of a special class of surface gravity wave groups, namely bichromatic waves. The model takes only two modes into account, namely the primary mode and the third order mode which is known to be most relevant for bichromatic waves with small frequency difference. Given an initial condition, an analytical expression for the maximal amplitude of the evolution of this initial wave group according to the model can be readily obtained. The aim of this investigation is to predict the amplification factor defined as the quotient between the maximal amplitude over all time & space and the initial maximal amplitude. Although this is a problem of general interest, as a case study, initial conditions in the form of a bichromatic wave group are taken. Using the low dimensional model it is found that the least upper bound of the maximal amplification factor for this bichromatic wave group is √2. To validate the analytical results of this model, a numerical simulation on the full model is also performed. As can be expected, good agreement is observed between analytical and numerical solutions for a certain range of parameters; when the initial amplitude is not too large, or when the difference of frequency is not too small. The results are relevant and motivated for the generation of waves in hydrodynamic laboratories.
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