physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...

, droplet-shaped waves are casual localized solutions of the

wave equation The (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields — as they occur in classical physics — such as mechanical waves (e.g. water waves, sound waves and s ...

closely related to the X-shaped waves, but, in contrast, possessing a finite

support Support may refer to: Arts, entertainment, and media * Supporting character Business and finance * Support (technical analysis) * Child support * Customer support * Income Support Construction * Support (structure), or lateral support, a ...

. 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 A tachyon () or tachyonic particle is a hypothetical particle that always travels faster than light. Physicists believe that faster-than-light particles cannot exist because they are not consistent with the known laws of physics. If such partic ...

) at infinite rectilinear motion to the case of a line source pulse started at time . The pulse front is supposed to propagate with a constant superluminal velocity (here is the speed of light, so ). In the cylindrical spacetime coordinate system , 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 
J(\tau ,z) H(\beta \tau -z) H(z),

where and are, correspondingly, the

Dirac delta In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire ...

and Heaviside step functions while is an arbitrary continuous function representing the pulse shape. Notably, for , so for as well. As far as the wave source does not exist prior to the moment , a one-time application of the

causality principle Causality is the relationship between causes and effects. While causality is also a topic studied from the perspectives of philosophy and physics, it is operationalized so that causes of an event must be in the past light cone of the event and ...

implies zero wavefunction for negative values of time. As a consequence, is uniquely defined by the problem for the wave equation with the time-asymmetric homogeneous initial condition :

\psi(\tau,\rho,z) = s(\tau,\rho,z) \\ & \psi(\tau,\rho,z) = 0 \quad \text \quad \tau < 0 \end

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.A.B. Utkin,
Droplet-shaped waves: casual finite-support analogs of X-shaped waves.
''arxiv.org'' 1110.3494 hysics.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), 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)

References

Wave mechanics

See also

References