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Droplet-shaped wave

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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 ( τ , ρ , z ) = δ ( ρ ) 2 π ρ J ( τ , z ) H ( β τ − z ) H ( z ) , {\displaystyle 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

[ ∂ τ 2 − ρ − 1 ∂ ρ ( ρ ∂ ρ ) − ∂ z 2 ] ψ ( τ , ρ , z ) = s ( τ , ρ , z ) ψ ( τ , ρ , z ) = 0 for τ < 0 {\displaystyle {\begin{aligned}&\left\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.

See also

References

  1. Recami, Erasmo (2004). "Localized X-shaped field generated by a superluminal electric charge" (PDF). Physical Review E. 69 (2): 027602. arXiv:physics/0210047. Bibcode:2004PhRvE..69b7602R. doi:10.1103/physreve.69.027602. PMID 14995594. S2CID 14699197.
  2. A.B. Utkin, Droplet-shaped waves: casual finite-support analogs of X-shaped waves. arxiv.org 1110.3494 (2011).
  3. A.B. Utkin, 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
  4. 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, ISBN 978-3-527-41195-5, pp. 287-306 (2013)
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