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

TL;DR

This paper investigates the second harmonic generation in surface water waves through experiments and compares the results with predictions from Stokes expansion and generalized NLS models, highlighting the importance of dissipation.

Contribution

It introduces experimental measurements of harmonic amplitudes and evaluates the accuracy of Stokes and generalized NLS predictions for water wave harmonics.

Findings

Dissipative NLS models outperform conservative ones in predictions.

Reasonable agreement between experiments and theoretical models.

Predictions from different methods show qualitative differences.

Abstract

When a mechanical wavemaker at one end of a water-wave tank oscillates with a frequency, $\omega_0$, time series of downstream surface waves typically include the dominant frequency (or first harmonic), $\omega_0$, along with the second, $2\omega_0$; third, $3\omega_0$; and higher harmonics. 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 Langmuir waves in plasmas. 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 measurements of the first harmonic band. We test the accuracy of these predictions by making two types of comparisons with the 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.