Abstract
The Green function appropriate to the wave equation with sources rotating about a fixed axis may readily be determined from the Liénard-Wiechert potential for a moving point source; but the resulting formula involves a sum over the roots of a transcendental equation, and thus appears at first sight to be unsuitable for analytical study. This paper presents a technique (‘the parametric method’) which bypasses the need to solve the transcendental equation, and leads to simple analytical expressions for the location in space of surfaces of constant field strength. These display a rich structure of spirals, folds, self-intersections, and lines of cusps. Parametric formulae are obtained for sections of the surfaces by various planes and cylinders; the resulting curves provide a convenient means of visualizing the surfaces in three dimensions, and diagrams are given of several families of such curves. They reveal the way in which the field produced by sources moving faster than the speed of sound (or light) differs from that produced at lower speeds; and the transition through exactly sonic (or ‘luminal') speeds is made particularly clear. The results are useful in the theory of transonic acoustics and Cherenkov radiation.
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Schedule a callMore From: Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences
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