Budden's Paradox Resolved

TL;DR

This paper resolves Budden's energy nonconservation paradox by showing that the pole approach introduces a localized dissipation term, ensuring energy balance regardless of wave direction, except for a special reflectionless case.

Contribution

It demonstrates that the paradox is resolved by interpreting the pole approach as a combination of a principal value and a delta function, providing the missing dissipation.

Findings

The pole approach yields a real principal value plus a delta function term.

The delta function supplies the necessary localized dissipation.

Reflectionless downward wave incidence remains a unique challenge.

Abstract

Budden's energy nonconservation paradox is dispelled herein by recognizing that pole approach to the spatial origin from below in the complex plane can be resolved into a real principal value plus $-i\pi$ times a Dirac delta, the imaginary coefficient whereof supplies just the right amount of localized dissipation to equilibrate the energy budget precisely, regardless of whether wave incidence be up or down. Only the reflectionless downward wave incidence remains as a counterintuitive challenge to physical anticipation, but at least a challenge liberated from its former onus of energy nonconservation.