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

Abstract

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.

Appendix A: Spectral peaks and means

Figures 7, 8, 9 and 10 contain plots of the evolution of the dimensional spectral peaks and dimensional spectral means for both the first and second harmonic bands for all four experiments. The spectral peak values were determined by selecting the frequency corresponding to the Fourier mode with largest magnitude at each gauge (for the experiments) or at each $\chi$-step (for the envelope equations). The (dimensional) spectral mean was computed using

Fig. 7

Plots of the spectral peaks (upper plots) and spectral means (lower plots) for the first harmonic band (left plots) and second harmonic band (right plots) versus distance down the tank for Experiment A

Fig. 8

Plots of the spectral peaks (upper plots) and spectral means (lower plots) for the first harmonic band (left plots) and second harmonic band (right plots) versus distance down the tank for Experiment B

Fig. 9

Plots of the spectral peaks (upper plots) and spectral means (lower plots) for the first harmonic band (left plots) and second harmonic band (right plots) versus distance down the tank for Experiment C

Fig. 10

Plots of the spectral peaks (upper plots) and spectral means (lower plots) for the first harmonic band (left plots) and second harmonic band (right plots) versus distance down the tank for Experiment D

$$\begin{aligned} \omega _m=\omega _0+\frac{\mathcal {P}}{\mathcal {M}}, \end{aligned}$$

(A.1)

where the factor of $\omega _0$ is necessary to take into account the fact that NLS-like models factor out the first harmonic, see Eq. (5). The quantities $\mathcal {P}$ and $\mathcal {M}$ were computed via

$$\begin{aligned}&\mathcal {M}=\sum _{j\in \mathbbm {B}}|a_j|^2, \end{aligned}$$

(A.2a)

$$\begin{aligned}&\mathcal {P}=\sum _{j\in \mathbbm {B}}\frac{2\pi j}{L}|a_j|^2, \end{aligned}$$

(A.2b)

where $\mathbbm {B}$ represents the set of frequencies in the band under consideration. The plots for the spectral peaks and means for the first harmonic band were first shown in Carter et al. [5]. However, the plots shown here are slightly different than those because they include surface tension effects while those in Carter et al. [5] did not.

This series of plots show that none of the envelope equations accurately models the evolution of the spectral peaks or means for all four experiments. This means that none of these models consistently models frequency downshift in the spectral peak or mean sense in an accurate manner.