Английская Википедия: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 [1] to the case of a line source pulse started at time Шаблон:Math. The pulse front is supposed to propagate with a constant superluminal velocity Шаблон:Math (here Шаблон:Math is the speed of light, so Шаблон:Math).
In the cylindrical spacetime coordinate system Шаблон:Math, 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
- <math>
s(\tau ,\rho ,z) = \frac{\delta (\rho )} {2\pi \rho} J(\tau ,z) H(\beta \tau -z) H(z), </math>
where Шаблон:Math and Шаблон:Math are, correspondingly, the Dirac delta and Heaviside step functions while Шаблон:Math is an arbitrary continuous function representing the pulse shape. Notably, Шаблон:Math for Шаблон:Math, so Шаблон:Math for Шаблон:Math as well.
As far as the wave source does not exist prior to the moment Шаблон:Math, a one-time application of the causality principle implies zero wavefunction Шаблон:Math for negative values of time.
As a consequence, Шаблон:Math is uniquely defined by the problem for the wave equation with the time-asymmetric homogeneous initial condition
- <math>\begin{align}
& \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}</math>
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]
See also
References
- ↑ Шаблон:Cite journal
- ↑ A.B. Utkin, Droplet-shaped waves: casual finite-support analogs of X-shaped waves. arxiv.org 1110.3494 [physics.optics] (2011).
- ↑ A.B. Utkin, Droplet-shaped waves: casual finite-support analogs of X-shaped waves. J. Opt. Soc. Am. A 29(4), 457-462 (2012), Шаблон:Doi
- ↑ 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, Шаблон:ISBN, pp. 287-306 (2013)
