Inhomogeneous Wave Equation and Weber’s Electrodynamics

It is well known that the inhomogeneous electric wave equation can be derived from Maxwell’s equations. On the other hand, it has been a common view that Weber’s electrodynamics—an alternative to Maxwell–Lorentz electromagnetism—is unable to produce a wave equation capable of explaining energy radiation. Recently, by introducing the concept of a polarizable vacuum, both longitudinal and transverse wave equations have been derived from Weber’s electrodynamics. However, these derivations did not incorporate source terms. In this paper, we attempt to include source terms, thereby deriving an inhomogeneous wave equation from Weber’s electrodynamics. Remarkably, this inhomogeneous wave equation takes the same form as the one derived from Maxwell’s equations. Although the equation itself permits longitudinal wave propagation, the requirement of charge conservation restricts the solution in vacuum to exclude longitudinal waves. Nevertheless, longitudinal waves can exist in the case of conduction currents (i.e., the movement of free charges) in electric wires.