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

TL;DR

This paper extends a model of superluminal wave generation to a causal scenario with a finite-duration source pulse, resulting in droplet-shaped waves that expand and tend toward X-shaped waves, providing a more realistic understanding of such phenomena.

Contribution

It introduces a causal model of superluminal wave generation with finite support, deriving analytical solutions for droplet-shaped waves that evolve into X-shaped waves over time.

Findings

Analytical solution for droplet-shaped waves with finite support

Droplet-shaped waves expand and tend to X-shaped waves

Model applicable to realistic superluminal source scenarios

Abstract

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.