Observation of Wave Turbulence in Vibrating Plates

The nonlinear interaction of waves in a driven medium may lead to wave turbulence, a state such that energy is transferred from large to small length scales. Here, wave turbulence is observed in experiments on a vibrating plate. The frequency power spectra of the normal velocity of the plate may be rescaled on a single curve, with power-law behaviors that are incompatible with the weak turbulence theory of Düring et al. [Phys. Rev. Lett. 97, 025503 (2006)]. Alternative scenarios are suggested to account for this discrepancy—in particular the occurrence of wave breaking at high frequencies. Finally, the statistics of velocity increments do not display an intermittent behavior.

Figure 1

The experiment. (a) Setup with a steel plate of dimensions $2\mathrm{m}\times 1\mathrm{m}$ and thickness $h=0.5\mathrm{mm}$; close-up view of the fixation. (b), (c) time series of the local transverse velocity measurements $v(t)$ for the forcing frequency $f_i=20\mathrm{Hz}$; duration of 10 times (b) and 1 time (c) the forcing period.

Figure 2

Power spectra of the transverse velocity. (a) raw $P_v(f)$ as a function of frequency $f$, for different values of injected power $I$ (in increasing order as displayed by the arrow: $<1\mathrm{mW}$, 2.3, 8.8, 26.4, 68.8, 136 mW); errors $\Delta f=2\mathrm{Hz}$ and $\Delta P_v={10}^{-7}(\mathrm{mm}/\mathrm{s}{)}^2/\mathrm{Hz}$. (b) Rescaled spectra $(I_0/I)P_v$ according to Eq. (2) vs $f/f_c$ for all forcing amplitudes, where $f_c$ is defined by $(I_0/I{)}^{1/2}{P}_v(f_c)={10}^{-5}{\mathrm{mm}}^2/\mathrm{s}$. Inset (c) evolution of $f_c$ with the forcing intensity. The continuous line is the best power law given in Eq. (2), yielding an exponent $\alpha =0.33$.

Figure 3

Damping factor for the present plate, from 27.

Figure 4

Statistical properties of the velocity increments ${\Delta }_{\tau }v=v(t+\tau )-v(t)$ (injected power 136 mW). Probability density functions compared to Gaussians in (a). Structure functions of order $p=2$, 3, 4, 5, 6, versus (b): the time lag $\tau$, and (c): the order 2 structure function $S_2$. Continuous lines are best power laws fits with exponents ${\zeta }_p$ (see text).