Frequency properties.

We examine the frequency properties of waves by performing the time-Fourier transform at each fixed wavenumber. A typical plot, for ${\bf k} = (17, 0)$, is shown in Figures 6. Our first observation is that we always see two peaks - the bigger one at the linear frequency and a smaller peak at a shifted frequency. We interpret the second peak as a nonlinear effect since there is no frequency shift in the linear system.

Figure 6: Frequency distribution of waveaction at a fixed wavenumber ${\bf k} = (17, 0)$.

[width=10cm]freq_k17.eps

Also, it appears that the measured ratio of squares of the peak frequencies is approximately equal to 2 for all wavenumbers (within 10% accuracy). This can be explained by the nonlinear term in the canonical transformation (9), e.g. $0(\epsilon)$-term which is quadratic with respect to the wave amplitude. In particular, the mode ${\bf k}/2$ makes contribution to this term which oscillates at frequency $2 \omega({\bf k}/2) = \sqrt{2 k}$ which appears to coincide with the second peak's frequency. Thus we see that contribution of ${\bf k}/2$ dominates in the nonlinear term of the canonical transformation.

The two-frequency character at each wavenumber has an interesting relation to the amplitude and phase dynamics as will be seen in the next section.