Droplet-shaped waves: causal finite-support analogs of X-shaped waves

J Opt Soc Am A Opt Image Sci Vis. 2012 Apr 1;29(4):457-62. doi: 10.1364/JOSAA.29.000457.


A model of steady-state X-shaped wave generation by a superluminal (supersonic) pointlike source infinitely moving along a straight line is extended to a more realistic causal scenario of a source pulse launched at time zero and propagating rectilinearly at a constant superluminal speed. In the case of an infinitely short (delta) pulse, the new model yields an analytical solution, corresponding to the propagation-invariant X-shaped wave clipped by a droplet-shaped support, which perpetually expands along the propagation and transversal directions, thus tending the droplet-shaped wave to the X-shaped one.