title: Droplet-shaped wave
text: # 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 = 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
  
      $s(\tau ,\rho ,z)={\frac {\delta (\rho )}{2\pi \rho }}J(\tau ,z)H(\beta \tau -z)H(z),$
  
  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
  
      ${\begin{aligned}&\left[\partial _{\tau }^{2}-\rho ^{-1}\partial _{\rho }(\rho \partial _{\rho })-\partial _{z}^{2}\right]\psi (\tau ,\rho ,z)=s(\tau ,\rho ,z)\\&\psi (\tau ,\rho ,z)=0\quad {\text{for}}\quad \tau <0\end{aligned}}$
  
  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.
  
  