Space-Time Resolved Wave Turbulence in a Vibrating Plate

Wave turbulence in a thin elastic plate is experimentally investigated. By using a Fourier transform profilometry technique, the deformation field of the plate surface is measured simultaneously in time and space. This enables us to compute the wave-vector-frequency ($\mathbf{k}$, $\omega$) Fourier spectrum of the full space-time deformation velocity. In the 3D ($\mathbf{k}$, $\omega$) space, we show that the energy of the motion is concentrated on a 2D surface that represents a nonlinear dispersion relation. This nonlinear dispersion relation is close to the linear dispersion relation. This validates the usual wave-number-frequency change of variables used in many experimental studies of wave turbulence. The deviation from the linear dispersion, which increases with the input power of the forcing, is attributed to weak nonlinear effects. Our technique opens the way for many new extensive quantitative comparisons between theory and experiments of wave turbulence.

Figure 1

(a) Sketch of the wave turbulence experiment. The specifically designed 2D mechanical system is made of a $2\times 1{\mathrm{m}}^2$, 0.4 mm thick stainless steel plate held vertically and set in motion by an electromagnetic vibrator at 30 Hz. The Fourier transform profilometry is based on the projection of a sine intensity pattern by a high definition video projector. The deformed image is then recorded by a high speed camera (1300 or 2600 fps). (b) Example of measurement of the deformation velocity on a 63 cm by 62 cm area.

Figure 2

(a) Space-time spectrum $E(\mathbf{k},\omega )$ of the deformation velocity (colors are log scaled). The cuts are located at $k_x=0$, $k_y=0$, and $\omega =5000\mathrm{rad}/\mathrm{s}$. The energy is localized on a surface in the ($\mathbf{k}$, $\omega$) space which confirms that the turbulent motion is due to a nonlinear superposition of waves following a dispersion relation. (b) Spectrum $E(k,\omega )$ obtained from $E(\mathbf{k},\omega )$ by integrating over the direction of $\mathbf{k}$. Inset: space spectrum $E(\mathbf{k})$ computed from $E(\mathbf{k},\omega )$ by summing over the frequencies. Contours are log scaled in both plots.

Figure 3

Nonlinear dispersion relation $\omega (k)$ computed from the line of maximum energy in the space-time spectrum of the plate deformation velocity for various input power $P$ of the forcing (from bottom to top $P^{1/2}=1$, 2, 3, 4 in arbitrary units). The dashed line is the linear dispersion relation (3). A systematic shift is observed which increases with the forcing power. The top inset shows the deviation from the linear dispersion. At high wave numbers, the shift is seen to be independent of $k$. The bottom inset shows the shift normalized by $P^{1/2}$.

Figure 4

Test of the validity of the change of variable $k\leftrightarrow \omega$ using the dispersion relation. Solid blue or dark gray line: direct measurement of the time spectrum $E(\omega )$—solid red or gray (respectively black dashed) line: $E(\omega )$ computed from the space spectrum $E(k)$ using the nonlinear (respectively linear) dispersion relation to change variable. Inset: same for the space spectrum $E(k)$. Solid blue or dark gray line: direct estimation of $E(k)$—red or gray (respectively black dashed) line: $E(k)$ computed from $E(\omega )$ by the change of variable using the nonlinear (respectively linear) dispersion relation.