Droplet-shaped wave
In physics, droplet-shaped waves are casual localized solutions of the wave equation closely related to the X-shaped waves, but, in contrast, possessing a finite support.
A family of the droplet-shaped waves was obtained by extension of the "toy model" of X-wave
generation by a superluminal point electric charge at infinite rectilinear motion
to the case of a line source pulse started at time. The pulse front is supposed to propagate
with a constant superluminal velocity .
In the cylindrical spacetime coordinate system,
originated in the point of pulse generation and oriented along the line of source propagation,
the general expression for such a source pulse takes the form
where and are, correspondingly,
the Dirac delta and Heaviside step functions
while is an arbitrary continuous function representing the pulse shape.
Notably,
for, so
for as well.
As far as the wave source does not exist prior to the moment,
a one-time application of the causality principle implies zero wavefunction
for negative values of time.
As a consequence, is uniquely defined by the problem for the wave equation with
the time-asymmetric homogeneous initial condition
The general integral solution for the resulting waves and the analytical description of their finite,
droplet-shaped support can be obtained from the above problem using the
STTD technique.