Elastic weak turbulence: From the vibrating plate to the drum

Abstract

Weak wave turbulence has been observed on a thin elastic plate in previous work. Here we report theoretical, experimental and numerical studies of wave turbulence in a 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 start from the 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.

Direct Evidence of a Dual Cascade in Gravitational Wave Turbulence

“…WT is a state dominated by waves of weak amplitude; all non-wave initial disturbances eventually die out. Such a system is characterized by a very specific ω-k spectrum that concentrates near the dispersion relation curve of the wave in question [17,18,21,23,30,32,39,41,44]. It is also the case for GW turbulence as we can see in Fig.…”

“…WT is of interest to many physical systems for which theoretical predictions have been made and verified numerically or experimentally. We have, among others, capillary waves [7][8][9][10][11][12][13][14] and gravity waves [15][16][17][18] on fluid surfaces, inertial waves in rotating hydrodynamics [19][20][21][22][23][24][25][26], elastic waves on thin vibrating plates [27][28][29][30][31][32], optical waves in optical fibers [33,34], waves in Bose-Einstein condensate [35,36], Kelvin waves on quantum vortex filaments [37][38][39], magnetostrophic waves in geodynamo [40,41] and magnetohydrodynamic waves in space plasmas [42][43][44][45][46][47]. Recently, a theory of WT has been developed for gravitational waves (GW) [48], a few years after their first direct detection [49].…”

“…We can also mention a strong similarity to elastic wave turbulence in the high tension limit. Indeed, both problems involve four-wave interactions with an inverse cascade of wave action [32]. This similarity could be a motivation to pursue the comparison in an analog laboratory experiment to better understand strong GW turbulence in cosmology and the formation of a metric condensate.…”

Wave turbulence: the case of capillary waves

“…Wave turbulence is a very common natural phenomenon found, for example, with/in gravity waves [6][7][8][9], capillary waves [see also the discussion in section IX, [10][11][12], quantum turbulence [13][14][15][16], nonlinear optics [17][18][19], inertial waves [20][21][22][23][24], magnetostrophic waves [25,26], elastic plates [27][28][29][30][31][32], plasma waves [33][34][35][36][37], or primordial gravitational waves [38][39][40]. These few examples demonstrate the vitality of the domain.…”

Saturation of the inverse cascade in surface gravity wave turbulence

“…Instead, a saturation is observed resulting from the emergence of highly dissipative nonlinear structures. Although well understood in 2D hydrodynamic turbulence [20] or Bose-Einstein condensation [21,22], such large-scales dynamics resulting from an inverse cascade is far from being fully understood for wave turbulence systems beyond gravity surface waves [7], such as for optical waves [8], waves in superfluid [9], plasma waves [23], or elastic waves [24,25].…”