Fig. 4: Time-resolved measurement.
a Schematic illustration of the wave propagation after exciting a Euclidean (top) and hyperbolic (bottom) drum with a short and spatially localized pulse. The waves travel along geodesics originating from the source (red lines) and wave fronts at different times are given by concentric circles perpendicular to the geodesics. Several equidistant circles with radii 0.5, 1, … (in the appropriate metric) are shown (black circles) for both cases, illustrating distances dE and dH to the source. b Broadband excitation pulse (blue) which is fed as a current pulse into node 31 at the boundary, and the voltage response measured at the same node (orange). The time corresponding to the instantaneous phases in d–f is marked by a red vertical line. c Frequency spectrum (blue) of the excitation pulse shown in b, demonstrating the wide range of frequencies contained in the pulse by comparison to the impedance to ground \(\left|{{Z}}_{31}\right|\) (gray; shown on a logarithmic scale on the right axis from 20 to 500 Ω). d Instantaneous phases of the pulse propagating on the hyperbolic drum (see legend) at time t = 2.032 μs. The nodes are indicated by black dots, and concentric hyperbolic circles with center at node 31 are shown in black to illustrate the hyperbolic metric. e Difference of the instantaneous phase φ at each node to the one at the source of the signal (node 31) φ0 vs. the hyperbolic distance dH to the source. f Difference of the instantaneous phase φ at each node to the one at the source of the signal (node 31) φ0 vs. the Euclidean distance dE to the the source. The shaded region in e, f indicates the approximate spread of the instantaneous phase as a function of dH and dE, respectively.
