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| In [[physics]], '''Droplet-shaped waves''' are casual localized solutions of the [[wave equation]] closely related to the [[X-wave|X-shaped waves]], but, in contrast, possessing a finite [[Support (mathematics)|support]].
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| A family of the droplet-shaped waves was obtained by extension of the "toy model" of [[X-wave]]
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| generation by a superluminal point electric charge ([[tachyon]]) at infinite rectilinear motion
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| <ref name=Recami2004>E. Recami, M. Zamboni-Rached and C. Dartora,
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| [http://pre.aps.org/abstract/PRE/v69/i2/e027602 Localized X-shaped field generated by a superluminal electric charge.]
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| ''Physical Review E'' '''69'''(2), 027602 (2004), doi:10.1103/PhysRevE.69.027602
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| </ref>
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| to the case of a line source pulse started at time {{math| ''t'' {{=}} 0}}. The pulse front is supposed to propagate
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| with a constant superluminal velocity {{math| ''v'' {{=}} ''βc''}} (here {{math| ''c''}} is the speed of light,
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| so {{math| ''β'' > 1}}).
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| In the cylindrical spacetime coordinate system {{math| ''τ{{=}}ct, ρ, φ, z''}},
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| originated in the point of pulse generation and oriented along the (given) line of source propagation (direction ''z''),
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| the general expression for such a source pulse takes the form
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| :<math>
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| s(\tau ,\rho ,z) =
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| \frac{\delta (\rho )} {2\pi \rho}
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| J(\tau ,z) H(\beta \tau -z) H(z),
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| </math> | |
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| where {{math|''δ''(•)}} and {{math|''H''(•)}} are, correspondingly,
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| the [[Dirac delta function|Dirac delta]] and [[Heaviside step function|Heaviside step]] functions
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| while {{math|''J''(''τ'', ''z'')}} is an arbitrary continuous function representing the pulse shape.
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| Notably,
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| {{math|''H'' (''βτ'' - ''z'') ''H'' (''z'') {{=}} 0}} for {{math|''τ'' < 0}}, so
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| {{math|''s'' (''τ'', ''ρ'', ''z'') {{=}} 0}} for {{math|''τ'' < 0}} as well.
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| As far as the wave source does not exist prior to the moment {{math|''τ'' {{=}} 0}},
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| a one-time application of the [[Causality (physics)|causality principle]] implies zero wavefunction
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| {{math|''ψ'' (''τ'', ''ρ'', ''z'')}} for negative values of time.
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| As a consequence, {{math|''ψ''}} is uniquely defined by the problem for the wave equation with
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| the time-asymmetric homogeneous initial condition
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| :<math>\begin{align}
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| & \left[
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| {\partial _\tau ^2 - {\rho ^{-1}}{\partial _\rho }\left( {\rho {\partial _\rho }} \right) - \partial _z^2}
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| \right]
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| \psi \left( \tau, \rho ,z \right)
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| =
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| s \left( \tau, \rho ,z \right)\\
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| & \psi \left( \tau, \rho ,z \right) = 0 \quad \mathrm{for} \quad \tau < 0
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| \end{align}</math>
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| The general integral solution for the resulting waves and the analytical description of their finite,
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| droplet-shaped support can be obtained from the above problem using the
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| [[spacetime triangle diagram technique|STTD technique]]
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| ,<ref name=Utkin2011>A.B. Utkin,
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| [http://arxiv.org/abs/1110.3494 Droplet-shaped waves: casual finite-support analogs of X-shaped waves.]
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| ''arxiv.org'' 1110.3494 [physics.optics] (2011).</ref>
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| ,<ref name=Utkin2012>A.B. Utkin,
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| [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</ref>
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| <ref name=Utkin2013>A.B. Utkin,
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| ''Localized Waves Emanated by Pulsed Sources: The Riemann-Volterra Approach''.
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| In: Hugo E. Hernández-Figueroa, Erasmo Recami, and Michel Zamboni-Rached (eds.)
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| [http://books.google.pt/books?isbn=3527671536 Non-diffracting Waves.] Wiley-VCH: Berlin,
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| <nowiki>ISBN 978-3-527-41195-5</nowiki>, pp. 287-306 (2013)</ref>
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| == See also ==
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| * [[X-wave]]
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| == References==
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| <references/>
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| [[Category:Wave mechanics]]
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Hi there! :) My name is Dominik, I'm a student studying Social Science Education from Su?Ureyri, Iceland.
Feel free to visit my webpage; Magnetic Bottle Opener