Fig. 3
From: Emergence of traveling waves in linear arrays of electromechanical oscillators
Minimal model of traveling wave synchronization. a Charge q and position h of an idealized oscillator as a function of its phase φ. b Phase-averaged electrostatic interaction between two weakly-coupled oscillators as a function of their phase difference χ. The interaction is scaled by \(f_{\mathrm{o}} = q_{\mathrm{o}}^2/\varepsilon W^2L\gamma\); the oscillator spacing is W = 0.12L. c Stable stationary solution for the phase difference χn of N = 15 oscillators. Experimental data for the same conditions is reproduced for comparison. The predicted phase differences are also plotted in (b) to show their relationship with the interaction function f( ). d Sine of the oscillator phase φn showing the wave-like pattern. e Dynamics of the N = 15 oscillators starting from random initial conditions. f Oscillator dynamics for N = 20 showing the periodic breaking of the traveling waves. In (c)–(e), the critical oscillator number is chosen to be N* = 15.5 as in experiment, which implies a frequency gradient Δ = 0.0096fo. The natural frequency is \(\omega _o = {\textstyle{3 \over {2\pi ^2}}}(W/a)^2f_o = 1.4f_o\) where W/a = 3 as in experiment; the ratio Δ/ωo = 0.0070 implies an electrode angle of θ = 1.1°
