Stationary phase approximation for the Mach surface of superluminally moving source

TL;DR

This paper develops a Fourier series method to analyze the potentials of superluminally rotating charges, deriving the Mach surface equation through asymptotic expansion, advancing understanding of superluminal electromagnetic sources.

Contribution

It introduces a Fourier series approach to compute potentials without retarded position calculations and derives the Mach surface equation via asymptotic expansion.

Findings

Potential distribution calculated using Fourier series

Mach surface equation derived from asymptotic expansion

Analysis of potential characteristics based on the expansion

Abstract

Theoretical study of superluminal sources of electromagnetic radiation boosted after the discovery of Cherenkov-Vavilov radiation. Later, the way to create fictitious sources moving superluminally was suggested. Different approaches have been proposed for the research of the distribution of the potential and the fields radiated by the superluminally moving charges. The simplest idealized cases of uniform rectilinear motion of the charge and of the charge rotating with constant angular speed open opportunities of a detailed analysis of the fields and potentials. We use Fourier series to calculate the potential distribution of point charge rotating with constant speed. An obvious advantage of this approach is that one no longer needs to calculate the retarded positions of the charge. The number of the retarded positions depends on the observation point and increases as the ratio {\omega}{R_0}/c rises, where c is the speed of light, {\omega} is the rotation frequency, and {R_0} is the radius of the circle. We demonstrate that equation of Mach surface can be obtained basing on the asymptotic expansion of the potential. We analyze some characteristics of the potential basing on this asymptotic expansion.