Figure 1

Overview of system behavior with increasing $\epsilon$ at $\Omega =21\mathrm{Hz}$. Images of typical patterns (top) with corresponding temporal profiles of surface heights (bottom). The black lines of length 6.5 cm denote the locations where the temporal profiles were acquired. (a) $\epsilon <0.32$ square patterns are stable with rare defects. (b) $0.32<\epsilon <0.5$ OTP, patterns deform via oscillations within the lattice. (c) $\epsilon >0.66$ DMT. Defect numbers increase abruptly, breaking the pattern into domains. Lattice point motion increases in amplitude and becomes temporally irregular.Reuse & Permissions

Figure 2

The OTP as a phase instability. (a) The Fourier spectra, $\Delta u(f)$, are independent of $\epsilon$ for low $\epsilon$ (light lines, $\epsilon <0.32$). At higher values of $\epsilon$, vibration amplitudes increase within a narrow range of frequencies (dark lines). (Inset) Spectra of surface-wave heights of the same experiments are $\epsilon$-independent, indicating that the oscillations are a phase instability. (b) $A_{\mathrm{otp}}$ as a function of $\epsilon$. $A_{\mathrm{otp}}$ is defined as the integrated amplitude of the spectra in (a) after subtraction of the low $\epsilon$ background spectra [light lines in (a)]. $A_{\mathrm{otp}}>0$ defines the transition to OTP at $\epsilon ={\epsilon }_{\mathrm{otp}}$. Data in (a) and (b) are for $\Omega =21\mathrm{Hz}$. (c) The peak frequency, $f_{\mathrm{otp}}$, of $\Delta u(f)$ is finite at onset and increases linearly with $\epsilon$. Full (open) symbols denote measurements in the OTP (DMT phase). When normalized by $\Omega$, the oscillation frequency curves collapse. Error bars are representative for all measurements in (c).Reuse & Permissions

Figure 3

Characterization of the OTP lattice waves. Peaks appear for $\epsilon >{\epsilon }_{\mathrm{otp}}$ solely in the temporal spectra of $\nabla \times \mathbf{u}$ (a), while the temporal spectra of $\nabla ·\mathbf{u}$ (b) are unaffected. The $\nabla \times \mathbf{u}$ oscillations are associated with transverse waves. Light lines are measurements at the stable phase, $\epsilon =0.29$, 0.31. Dark lines are measurements after OTP onset, $\epsilon =0.45$, 0.47, 0.49. $\Omega =21\mathrm{Hz}$. (c) Mean temporal cross-correlation amplitudes of $\nabla \times \mathbf{u}$ at increasing distances $d$ along lattice lines ($\Omega =21\mathrm{Hz}$, $\epsilon =0.47$). The correlations peak at times linearly growing with $d$. The peak amplitudes decay exponentially with $d$. Cross correlations were obtained using time series whose duration was much longer than $1/f_{\mathrm{otp}}$, and results were averaged over all lattice points. Wave velocities (d) and attenuation lengths (e) as a function of $\epsilon$ and $\Omega$. Velocities (decay lengths) were extracted by linear (exponential) fits to the correlation peak time (amplitude) vs distance. Wave velocities were independent of both $\epsilon$ and $\Omega$.Reuse & Permissions

Figure 4

The transition to DMT and wave number selection. Phases, denoted by shading, are (from left to right): stable, OTP, and DMT. (a) Defect density (diamonds) and correlation time (circles) as a function of $\epsilon$. $\Omega =21\mathrm{Hz}$. Beyond $\epsilon ={\epsilon }_d$, defect density sharply increases. In parallel, correlation times drop sharply. This shows clearly that the defects are the source of both spatial and temporal disorder. Correlation times are defined by the times where the autocorrelation function of the surface elevation $h(x,y,t)$ equals $\frac{1}{2}$. Both correlation times and defect density show little change for $\epsilon <{\epsilon }_d$. (b) The linear growth of the instability vibration amplitude, $A_{\mathrm{otp}}$, terminates at $\epsilon ={\epsilon }_d$ with a sharp increase. $\Omega =21\mathrm{Hz}$. (c) The time-space-averaged wave number, $k$, as a function of $\epsilon$ for $\Omega =36\mathrm{Hz}$. $k$ decreases during both the stable and DMT phases and is nearly constant throughout the OTP.Reuse & Permissions