Space-Time Folding of the Wake Produced by a Supervelocity Rotating Point Source

Wave sources moving faster than the waves they emit create a wake whose topological features are directly related to the geometry of the source trajectory. These features can be understood by considering space-time surfaces representing past emitted wave fronts. Specifically, for a supervelocity source moving along a circular path the space-time envelope folds and a cusp appears on the inner part of the wake. As a result, the wake is ultimately contained within two parallel corotating spiraling branches. In this Letter we take advantage of the low phase speed of water waves to study experimentally supervelocity sources moving at velocities up to several time the wave speed. We image in real time their emission patterns and characterize the topological features of their wakes.

Figure 1

(a) and (b) Surfaces composed of the past wave fronts emitted by a source moving along a circular path at Mach numbers $\mathrm{Ma}=0.5$ and $\mathrm{Ma}=4$, respectively. Inner black lines show the space-time trajectories of the source. (c) and (d) are projections along the time direction of cones (a) and (b), respectively. Only wave fronts emitted periodically are represented. Thick black circles are the source trajectories. The dashed line marks the position of an observer. Wave fronts intersecting at this position are highlighted. Positions and directions of propagation of the associated images are given by the corresponding arrows.

Figure 2

(a) Schematic of the experimental setup. A pressure source is moving on a circular path over a tank filled with $h_0=1\mathrm{cm}$ of water. (b) Experimental profile of the cavity formed on the water surface (solid line) by a still and steady pressure source and profile (dashed line) computed from the Gaussian fit of the pressure profile. (c) Left axis. Phase ($c_{\varphi }$) and group ($c_g$) speed of the waves as a function of the wave number $k$ obtained from Eq. (3). Right axis. Weights (bracket term in Eq. (4) of the Bessel functions $J_n(kr)$ in terms of which we decompose the field. To each peak corresponds an order $n$ ranging from 1 to 11 (from left to right). Amplitudes are given in arbitrary units.

Figure 3

(a)–(c) Experimental wave fields produced on the water surface by a pressure source moving on a circular path at ${\mathrm{Ma}}_{\min }=1.2$, ${\mathrm{Ma}}_{\min }=2$ and ${\mathrm{Ma}}_{\min }=3.3$ respectively. (d)–(f) show corresponding wave fields computed from the hydrodynamic model. White oriented dashed circles depict the source trajectories while white arrows point at the current positions of the sources. 1.5 cm long white bars are superimposed on the fields for scale.

Figure 4

(a) Radius of the cusp trajectory $R_{\mathrm{cusp}}^{\exp }$ as a function of the inverse of the rotation pulsation $1/\mathrm{\Omega }$ for trajectory radii of $R=3\mathrm{cm}$ (blue diamonds) and $R=5\mathrm{cm}$ (black circles). Red crosses denote values obtained from the hydrodynamic model while the solid black line shows the analytical solution. (b) Angular distance $\alpha \mathrm{wake}$ between the two branches of the spiraling wake (inset) as a function of the Mach number ${\mathrm{Ma}}_{\min }$. The solid black line gives the analytical solution. (c) Wake propagation for a source moving along a straight line. (d) Corresponding schematic in the case of a source moving on a circular path.