Generation of a Bessel–Gauss pulse from a moving disk source distribution

Assuming a moving disk source distribution, the Riemann method and Gegenbauer's addition theorem in a mixed coordinate-space–phase-space formulation have been used to derive a zeroth-order Bessel–Gauss pulse solution to the inhomogeneous wave equation. In the region where τ2≳ρeff2+z2, the total solution can be separated into a wave that attenuates quickly and a Bessel–Gauss pulse that exists only after a certain time and continues for long times. In the region where z2⩽τ2≲ρeff2+z2, only a short-time solution exists.