Figure 5: Mode analysis of the wave field.
From: Self-organization into quantized eigenstates of a classical wave-driven particle
(a–d) Discretization of the radius of the smallest circular orbit (1,1). The radial force is dominated by the centred zero-order Bessel J0 function. The generated global wave field is h0(r)=A0(R)J0(kFr). (a) The amplitude A0(R) depends on the radius R of the trajectory: A0(R)∝J0(kFr). In b are shown three surface profiles hR(r) created by drops orbiting at three radii: R1 that correspond to the first zero of J0(kFr) and RA and RB that are slightly smaller and slightly larger, respectively (see a). Note that the circular orbit of radius R1 does not excite the J0 mode of the global wave field. (c) Whenever R does not coincide with a zero of the J0 mode, a mean wave field is generated so that the drop has an additional potential energy Ep(R)∝
(kFR). (d) If the radius is slightly smaller or larger than one of the radii Rn, the J0 mode is excited and exerts an additional ‘quantization’ force onto the droplet :−(∂Ep/∂R)∝J1(kFR)J0(kFR). (e) The experimental trajectory of an experimental circular orbit (n=1, m=1) at a memory parameter M=32. Scales are in λF units. (f) Spectral decomposition of the wave field in centred Bessel functions. The trajectory being close to the ideal, the amplitude of the J0 mode is weak, the dominant J1 mode is responsible for the azimuthal propulsion of the droplet. (g) An experimentally observed lemniscate trajectory (n=2, m=0). The latest M impacts are shown as open dots. It is superimposed on the reconstructed global wave field. (h) The spectral decomposition of this wave field showing that for this near-ideal orbit the J2 mode is specifically weak.
