Treatment of retardation effects in calculating the radiated electromagnetic fields from the lightning discharge

The analytical treatment of retardation effects in calculating lightning electromagnetic fields far from the source has often involved the use of a so-called F factor. The literature concerning the F factor in the lightning field equations is often confusing and sometimes in error. The aim of this paper is to clarify, to correct when needed, and to extend previous views of the F factor by considering and consolidating the various situations, both mathematically and physically, in which this factor can occur. The F factor arises because of the retardation effects occurring when the distance to the observer from a point on a propagating current is changing with time. In this paper we (1) discuss the various situations in which the F factor can arise, such as in the determination of the "radiating" channel length "seen" by the observer, in the calculation of fields due to a propagating current wave up or down the discharge channel, and in the calculation of fields due to a propagating current discontinuity extending the channel upward; (2) give a unifying physical interpretation for this factor; and (3) show that the retardation effects in calculating lightning fields can be accounted for without the explicit use of an F factor. Relative to item 2 above, we will show that for simple return-stroke models like the transmission line (TL) and traveling current source (TCS) models, in which the current at one point on the channel appears at another point at another time, the F factor associated with current behind the front can be interpreted physically as the ratio of the apparent propagation speed (upward or downward) of the current wave "seen" by a distant observer to its actual speed. In the TL model, a current wave moves upward at speed nu, the same speed as that of the front, and the F factor is given by [1-(nu/c) cos theta](-1), where theta is the angle between the direction of propagation of the source and the line joining the source point and the field point (observer), and c is the speed of light in vacuum. In the TCS model, a current wave moves downward at a speed equal c, and the F factor is given by [1+cos theta](-1). The F factor associated with an upward propagating current discontinuity can always (in any model) be interpreted physically as the ratio of the apparent propagation speed of the discontinuity to its actual speed and is given by the same expression as for the F factor for the upward propagating current wave in the TL model.