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.A.B. Utkin, Droplet-shaped waves: casual finite-support analogs of X-shaped waves. arxiv.org 1110.3494 [physics.optics] (2011).A.B. Utkin, Droplet-shaped waves: casual finite-support analogs of X-shaped waves. J. Opt. Soc. Am. A 29(4), 457-462 (2012), A.B. Utkin, Localized Waves Emanated by Pulsed Sources: The Riemann-Volterra Approach. In: Hugo E. Hernández-Figueroa, Erasmo Recami, and Michel Zamboni-Rached (eds.) Non-diffracting Waves. Wiley-VCH: Berlin, , pp. 287-306 (2013)
See also X-wave
References