Modeling the Second Harmonic in Surface Water Waves Using Generalizations of NLS

If a wavemaker at one end of a water-wave tank oscillates with a particular frequency, time series of downstream surface waves typically include that frequency along with its harmonics (integer multiples of the original frequency). This behavior is common for the propagation of weakly nonlinear waves with a narrow band of frequencies centered around the dominant frequency such as in the evolution of ocean swell, pulse propagation in optical fibers, and Bose-Einstein condensates. Presented herein are measurements of the amplitudes of the first and second harmonic bands from four surface water wave laboratory experiments. The Stokes expansion for small-amplitude surface water waves provides predictions for the amplitudes of the second and higher harmonics given the amplitude of the first harmonic. Similarly, the derivations of the NLS equation and its generalizations (models for the evolution of weakly nonlinear, narrow-banded waves) provide predictions for the second and third harmonic bands given amplitudes of the first harmonic band. We test the accuracy of these predictions by making two types of comparisons with experimental measurements. First, we consider the evolution of the second harmonic band while neglecting all other harmonic bands. Second, we use explicit Stokes and generalized NLS formulas to predict the evolution of the second harmonic band using the first harmonic data as input. Comparisons of both types show reasonable agreement, though predictions obtained from dissipative generalizations of NLS consistently outperform the conservative ones. Finally, we show that the predictions obtained from these two methods are qualitatively different.