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 = βc (here c is the speed of light, so β > 1).
In the cylindrical spacetime coordinate system τ=ct, ρ, φ, 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
where δ(•) and H(•) are, correspondingly, the Dirac delta and Heaviside step functions while J(τ, z) is an arbitrary continuous function representing the pulse shape. Notably, H (βτ − z) H (z) = 0 for τ < 0, so s (τ, ρ, z) = 0 for τ < 0 as well.
As far as the wave source does not exist prior to the moment τ = 0, a one-time application of the causality principle implies zero wavefunction ψ (τ, ρ, z) for negative values of time.
As a consequence, ψ 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.