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When a mechanical wavemaker at one end of a water-wave tank oscillates with afrequency, $\omega_0$, time series of downstream surface waves typicallyinclude the dominant frequency (or first harmonic), $\omega_0$, along with thesecond, $2\omega_0$; third, $3\omega_0$; and higher harmonics. This behavior iscommon for the propagation of weakly nonlinear waves with a narrow band offrequencies centered around the dominant frequency such as in the evolution ofocean swell, pulse propagation in optical fibers, and Langmuir waves inplasmas. Presented herein are measurements of the amplitudes of the first andsecond harmonic bands from four surface water wave laboratory experiments.  The Stokes expansion for small-amplitude surface water waves providespredictions for the amplitudes of the second and higher harmonics given theamplitude of the first harmonic. Similarly, the derivations of the NLS equationand its generalizations (models for the evolution of weakly nonlinear,narrow-banded waves) provide predictions for the second and third harmonicbands given measurements of the first harmonic band. We test the accuracy ofthese predictions by making two types of comparisons with the experimentalmeasurements. First, we consider the evolution of the second harmonic bandwhile neglecting all other harmonic bands. Second, we use explicit Stokes andgeneralized NLS formulas to predict the evolution of the second harmonic bandusing the first harmonic data as input. Comparisons of both types showreasonable agreement, though predictions obtained from dissipativegeneralizations of NLS consistently outperform the conservative ones. Finally,we show that the predictions obtained from these two methods are qualitativelydifferent.

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