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[1] to the case of a line source pulse started at time . The pulse front is supposed to propagatewith 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
s(\tau,\rho,z)=
| \delta(\rho) | |
| 2\pi\rho |
J(\tau,z)H(\beta\tau-z)H(z),
where and are, correspondingly, the Dirac delta and Heaviside step functionswhile 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 withthe time-asymmetric homogeneous initial condition
\begin{align} &\left[ \partial
| 2 | |
| \tau |
-\rho-1\partial\rho(\rho\partial\rho)-\partial
| 2 | |
| z |
\right] \psi(\tau,\rho,z)=s(\tau,\rho,z)\\ &\psi(\tau,\rho,z)=0 for \tau<0 \end{align}
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.[2] [3] [4]