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 $t = 0$. The pulse front is supposed to propagate with a constant superluminal velocity $v = &beta;c$ (here $c$ is the speed of light, so $&beta; > 1$).

In the cylindrical spacetime coordinate system $&tau;=ct, &rho;, &phi;, z$, 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) = \frac{\delta (\rho )} {2\pi \rho} J(\tau ,z) H(\beta \tau -z) H(z), $$

where $&delta;(&bull;)$ and $H(&bull;)$ are, correspondingly, the Dirac delta and Heaviside step functions while $J(&tau;, z)$ is an arbitrary continuous function representing the pulse shape. Notably, $H (&beta;&tau; − z) H (z) = 0$ for $&tau; < 0$, so $s (&tau;, &rho;, z) = 0$ for $&tau; < 0$ as well.

As far as the wave source does not exist prior to the moment $&tau; = 0$, a one-time application of the causality principle implies zero wavefunction $&psi; (&tau;, &rho;, z)$ for negative values of time.

As a consequence, $&psi;$ is uniquely defined by the problem for the wave equation with the time-asymmetric homogeneous initial condition


 * $$\begin{align}

& \left[ \partial _\tau ^2 - \rho^{-1} \partial_\rho (\rho \partial_\rho) - \partial _z^2 \right] \psi(\tau,\rho,z) = s(\tau,\rho,z) \\ & \psi(\tau,\rho,z) = 0 \quad \text{for} \quad \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.