Droplet-shaped waves: Causal finite-support analogs of X-shaped waves
  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 constant superluminal speed. In the case of 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. 
Andrei B. Utkin
physics.optics math-ph math.MP physics.class-ph