On the phenomenon of oscillations in the method of auxiliary sources

When one applies the method of auxiliary sources to scattering problems involving perfect conductors, one first seeks fictitious auxiliary currents located inside the conductor, and then determines the field from these currents. For a simple two-dimensional problem involving an infinite circular cylinder illuminated by an electric current filament, it has recently been shown analytically that it is possible to have divergent auxiliary currents (to make this statement precise, one must properly normalize the currents), together with a convergent field. It was also shown-through numerical investigations-that the aforementioned divergence appears as abnormal, rapid oscillations. In the present paper, we investigate such phenomena in more detail, with particular emphasis on oscillations. For a perfectly conducting ground plane illuminated by an electric current filament, we once again demonstrate the possibility of having divergent, oscillating currents producing a convergent field. We develop an asymptotic formula for the oscillating current values, which sheds light on the nature of the oscillations. We revisit the circular-cylinder problem to develop a similar asymptotic formula. We also discuss roundoff errors, and possible generalizations to scatterers of other shapes. The present study is to a great extent analytical, with the analytical predictions confirmed and supplemented by numerical results.