Dynamics of the wave turbulence spectrum in vibrating plates: A numerical investigation using a conservative finite difference scheme

Abstract

The dynamics of the local kinetic energy spectrum of an elastic plate vibrating in a wave turbulence (WT) regime is investigated with a finite difference, energy-conserving scheme. The numerical method allows the simulation of pointwise forcing together with realistic boundary conditions, a set-up which is close to experimental conditions. In the absence of damping, the framework of non-stationary wave turbulence is used. Numerical simulations show the presence of a front propagating to high frequencies, leaving a steady spectrum in its wake. Self-similar dynamics of the spectra are found with and without periodic external forcing. For the periodic forcing, the mean injected power is found to be constant, and the frequency at the cascade front evolves linearly with time resulting in a increase of the total energy. For the free turbulence, the energy contained in the cascade remains constant while the frequency front increases as t 1/3 . These self-similar solutions are found to be in accordance with the kinetic equation derived from the von Kármán plate equations. The effect of the pointwise forcing is observable and introduces a steeper slope at low frequencies, as compared to the unforced case. The presence of a realistic geometric imperfection of the plate is found to have no effect on the global properties of the spectra dynamics. The steeper slope brought by the external forcing is shown to be still observable in a more realistic case where damping is added.

Multi-scale simulation of nonlinear thin-shell sound with wave turbulence

“…The synthesized turbulence sound waveform is shown below. We note that this spectrum, while generated phenomenologically, agrees qualitatively with the prediction by nonlinear finite difference simulation, as shown in Figure 16 of [Ducceschi et al 2014]. …”

“…The resulting sound exhibits a wideband spectrum where linear modal vibrational frequencies can no longer be easily distinguished [Chaigne et al 2005]. Thin-plate wave turbulence has been extensively studied using the Föppl-von Kár-mán equation for large deflections [Ducceschi et al 2014;Düring et al 2006;Touzé et al 2012]. As the vibrational energy is transferred from the input energy scales to the high-frequency scales, the Kolmogorov-Zakharov spectrum emerges again.…”

String/frets contacts in the electric bass sound: Simulations and experiments

“…A more quantitative viewpoint on the energy transfer can be given by dening a characteristic frequency ν c as [29]:…”

Elastic weak turbulence: From the vibrating plate to the drum

“…Here we focus on a distinct physical system which provides a paradigm for studying the transition non dispersive wave turbulence by means of both experimental realizations and computational studies: the elastic plate under tension. The elastic plate is a rich physical system for wave turbulence which has been the object of a growing interest [15][16][17][18][19][20][21][22][23][24]. A shaken metal plate has been used for centuries to mimic the thunder noise in theaters [25] and in the first half of the 20th century as a reverberator for analog signal processing [26].…”