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 (tachyon) 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 (here is the speed of light,
so ).
In the cylindrical spacetime coordinate system ,
originated in the point of pulse generation and oriented along the (given) line of source propagation (direction "z"),
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.