Figure 1

(a) Schematic illustration of the experimental apparatus, a fluid bath driven vertically in a sinusoidal manner with amplitude $A_0$ and frequency $f$. (b) Faraday mode in a circular corral of radius $R=10.1$ mm and depth $h_0=5$ mm, driven at $f=80$ Hz, for which ${\gamma }_F=4.6$ and the Faraday wavelength is ${\lambda }_F=4.75$ mm. The frequency was selected to ensure that the wave field just above the Faraday threshold was stationary. Note that the bright rings correspond to areas of small surface slope, indicating local extrema of wave amplitude. (c)–(e) Typical trajectories of a walker with diameter $D=0.8$ mm with mean velocity $\overline{v}=11$ mm/s in the same circular corral at (c) low memory $\Gamma =0.18$ and intermediate memories (d) $\Gamma =0.1$ and (e) $\Gamma =0.06$.Reuse & Permissions

Figure 2

(a) Top view of a walker exploring a circular corral in the long-path-memory limit. Its complex pilot-wave field is apparent; its trajectory is indicated in white. (b) Oblique view of a walker and its pilot wave exploring a circular corral. The dashed line indicates the edge of the corral.Reuse & Permissions

Figure 3

(a) Trajectories of a droplet of diameter $D=0.67$ mm walking in a circular corral with radius $R=14.3$ mm and depth $h_0=6.6$ mm, driven at $f=70$ Hz, for which ${\gamma }_F=3.7$. Trajectories of increasing length in the long-path-memory limit ($\Gamma =0.011$) are color coded according to droplet speed (mm/s). (b) Probability distribution of the walking droplet's position.Reuse & Permissions

Figure 4

(a) Histogram of radial position, (b) velocity variation from the mean ($\overline{\left|v\right|}=8.66$ mm/s), and (c) radial dependence of the normalized radial velocity ($|v_R|/|v|$). In (a), the minima in the probability amplitude approximately correspond to maxima in the walker velocity, maxima in the normalized radial velocity, and zeros in the amplitude of the fundamental cavity mode (upper curve). The section A-B represents a radial slice of the cavity's Faraday mode, with bright bands indicating local extrema. Averaging windows and bin widths are fixed at $0.012R$. In (c), values of 1 and 0 correspond to purely radial and azimuthal motion, respectively. (d) Four sample trajectories extracted from the complete trajectory indicate a tendency to orbital motion along particular radii. Different colors serve only to demarcate different trajectories. In all plots, the dashed lines represent maxima in the amplitude of the fundamental cavity mode.Reuse & Permissions