Droplet-shaped wave

Inphysics,droplet-shaped wavesare casual localized solutions of thewave equationclosely related to theX-shaped waves,but, in contrast, possessing a finitesupport.

A family of the droplet-shaped waves was obtained by extension of the "toy model" ofX-wave generation by a superluminal point electric charge (tachyon) at infinite rectilinear motion ^[1] 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 (directionz), 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, theDirac deltaandHeaviside stepfunctions 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 thecausality principleimplies 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.^[2]^[3]^[4]

References

^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.
^A.B. Utkin, Droplet-shaped waves: casual finite-support analogs of X-shaped waves. arxiv.org1110.3494 [physics.optics] (2011).
^ A.B. Utkin, Droplet-shaped waves: casual finite-support analogs of X-shaped waves. J. Opt. Soc. Am. A29(4), 457-462 (2012),doi:10.1364/JOSAA.29.000457
^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)

[Recami2004-1] 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.

[Utkin2011-2] A.B. Utkin, Droplet-shaped waves: casual finite-support analogs of X-shaped waves. arxiv.org1110.3494 [physics.optics] (2011).

[Utkin2012-3] A.B. Utkin, Droplet-shaped waves: casual finite-support analogs of X-shaped waves. J. Opt. Soc. Am. A29(4), 457-462 (2012),doi:10.1364/JOSAA.29.000457

[Utkin2013-4] 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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See also

References