Elastic weak turbulence: From the vibrating plate to the drum

Weak wave turbulence has been observed on a thin elastic plates since the work by Düring et al. [Phys. Rev. Lett. 97, 025503 (2006)]. Here we report theoretical, experimental, and numerical studies of wave turbulence in a forced thin elastic plate submitted to increasing tension. When increasing the tension (or decreasing the bending stiffness of the plate) the plate evolves progressively from a plate into an elastic membrane as in drums. We first consider a thin plate and increase the tension in experiments and numerical simulations. We observe that the system remains in a state of weak turbulence of weakly dispersive waves. This observation is in contrast with what has been observed in water waves when decreasing the water depth, which also changes the waves from dispersive to weakly dispersive. The weak turbulence observed in the deep water case evolves into a solitonic regime. Here no such transition is observed for the stretched plate. We then apply the weak turbulence theory to the membrane case and show with numerical simulations that indeed the weak turbulence framework remains valid for the membrane and no formation of singular structures (shocks) should be expected in contrast with acoustic wave turbulence.

Figure 1

Experimental setup. A $1.9\times 1{\mathrm{m}}^2$, 0.5-mm-thick steel plate is stretched vertically. Vibrations are generated by an electromagnetic shaker at the bottom of the plate. The deformation is measured by the Fourier transform profilometry technique in which a pattern is projected on the plate and recorded by a high-speed camera (see text for details).

Figure 2

(a) Normalized spatiotemporal spectrum $E^v(k_x,0,\omega )/E^v(k_f,0,{\omega }_f)$ of the velocity field in the stretching direction for experiment E1 with $T_x=4$ kN/m and $T_y=0$. The crossover frequency ${\omega }_c/2\pi =290$ Hz is shown as a vertical dashed line. (b) Same dataset but the spectrum $E^v(0,k_y,\omega )/E^v(0,k_f,0,{\omega }_f)$ is shown. (c) $E^v(k_x,0,\omega )/E^v(k_f,0,{\omega }_f)$ for experiment E2 with a larger value of the tension $T_x=10$ kN/m and the maximum amplitude forcing permitted by the electromagnetic shaker. The crossover frequency ${\omega }_c/2\pi =725$ Hz is shown as a vertical dashed line. (d) $E^v(k_x,0,\omega )/E^v(k_f,0,{\omega }_f)$ for a numerical simulation N1 with $T_x=4$ kN/m and $C_{\exp }^2$ indented to reproduce qualitatively the experiment E1. In all cases the black line corresponds to the dispersion relation taking into account the tension (5) and the white line corresponds to the dispersion relation without tension (8).

Figure 3

Normalized spatiotemporal spectrum of the velocity field $E^v(k,\omega )/E^v(k_f,{\omega }_f)$ for the FvK-NS: (a) N2 with $C_{\exp }^2$, (b) N5, $C_{\exp }^2/10$, (c) N6, $C_{\exp }^2/100$, and (d) N7, $C_{\exp }^2/1000$. In all cases the black line corresponds to the dispersion relation (5) and the white line corresponds to the dispersion relation without tension (8). In (a) and (b) the vertical dashed line shows the crossover frequency ${\omega }_c$ [which is not in the displayed range for (c) and (d)].

Figure 4

Snapshots of the deformation of the plate for the FvK-NS: (a) N2 with $C_{\exp }^2$, (b) N5 with $C_{\exp }^2/10$, (c) N6 with $C_{\exp }^2/100$, and (d) N7 with $C_{\exp }^2/1000$.

Figure 5

$E^v\left(k\right)/E^v\left(k_f\right)$ for FvK-NS N2 to N7, i.e., for $C^2$ decreasing from $C_{\exp }^2$ to $C_{\exp }^2/1000$ with a constant tension. The slope of the deformation is kept roughly constant equal to about 2%. The circles correspond to the crossover wave number $k_c$ for the corresponding FvK-NS (when it is small than the dissipative cutoff $k_d=15k_f$). The bottom dashed line shows the theoretical scaling predicted by WTT applied to the pure membrane ($C^2=0$) $E^v\left(k\right)\propto k^{-4/3}$. The top dashed line shows the theoretical shape $k{log}^{1/3}(k_d/k)$ expected from the WTT applied to the plate [15].

Figure 6

Wave-number spectrum of the deformation $E^{\zeta }\left(k\right)$ for the high-resolution FvK-NS N9 of the pure membrane. The spectrum has been compensated either by $k^{10/3}$ (solid line), which is the WTT prediction for a membrane, or by $k^3$ (dashed line), corresponding to the WTT prediction for a plate with $C^2>0$ and $T=0$ (discarding the logarithmic corrections). Forcing acts in modes $k\in (1,4)$.

Figure 7

(a) Wave-number-frequency spectrum $E^{\zeta }$ spectrum of the height of the waves for the same moderate resolution simulation shown in Fig. 6. The black line is the linear dispersion. (b) Snapshot of the deformation of the membrane for the same simulation.

Figure 8

(a) FvK-NS N10 forced at small scale. The dashed line shows the $k^{-3}$ scaling expected from the WTT predictions. (b) Wave-number-frequency spectrum of the deformation for a moderate resolution simulation N11 forced at $k_f=200$ rad/m. The solid black line is the linear dispersion relation.

Figure 9

Evolution of the normalized spectral widening $\delta \omega /\omega$ as a function of $k$ for the pure membrane (FvK-NS N8). The dashed line is the WTT prediction $\delta \omega /\omega \propto k^{-2/3}$.

Figure 10

The resonant manifold for $k_2/p=.5$ (a), $k_2/p=1$ (b), $k_2/p=1.5$ (c), and $k_2/p=100$ (d). ${\theta }_p=\pi$ by choosing the system of coordinates.