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 {{math| t {{=}} 0}}. The pulse front is supposed to propagate
with a constant superluminal velocity {{math| v {{=}} βc}} (here {{math| c}} is the speed of light,
so {{math| β > 1}}).
In the cylindrical spacetime coordinate system {{math| 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 {{math|δ(•)}} and {{math|H(•)}} are, correspondingly,
the Dirac delta and Heaviside step functions
while {{math|J(τ, z)}} is an arbitrary continuous function representing the pulse shape.
Notably,
{{math|H (βτ − z) H (z) {{=}} 0}} for {{math|τ < 0}}, so
{{math|s (τ, ρ, z) {{=}} 0}} for {{math|τ < 0}} as well.
As far as the wave source does not exist prior to the moment {{math|τ {{=}} 0}},
a one-time application of the causality principle implies zero wavefunction
{{math|ψ (τ, ρ, z)}} for negative values of time.
As a consequence, {{math|ψ}} is uniquely defined by the problem for the wave equation with
the time-asymmetric homogeneous initial condition
:
& \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
[https://arxiv.org/abs/1110.3494 Droplet-shaped waves: casual finite-support analogs of X-shaped waves.]
arxiv.org 1110.3494 [physics.optics] (2011).
A.B. Utkin,
[http://www.opticsinfobase.org/josaa/abstract.cfm?uri=josaa-29-4-457 Droplet-shaped waves: casual finite-support analogs of X-shaped waves.]
J. Opt. Soc. Am. A 29(4), 457-462 (2012), {{doi|10.1364/JOSAA.29.000457}}
Localized Waves Emanated by Pulsed Sources: The Riemann-Volterra Approach.
In: Hugo E. Hernández-Figueroa, Erasmo Recami, and Michel Zamboni-Rached (eds.)
[https://books.google.com/books?isbn=3527671536 Non-diffracting Waves.] Wiley-VCH: Berlin, {{ISBN|978-3-527-41195-5}}, pp. 287-306 (2013)