1 Answers
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 at infinite rectilinear motionto the case of a line source pulse started at time t = 0. The pulse front is supposed to propagatewith a constant superluminal velocity v = βc.
In the cylindrical spacetime coordinate system τ=ct, ρ, φ, z,originated in the point of pulse generation and oriented along the line of source propagation ,the general expression for such a source pulse takes the form
where δ and H are, correspondingly, the Dirac delta and Heaviside step functionswhile J is an arbitrary continuous function representing the pulse shape.Notably, H H = 0 for τ < 0, so s = 0 for τ < 0 as well.