Validity of the f cos i Theorem for the Absorption of Very Low Frequency Radiowaves in the Ionosphere

Abstract

MARTYN¹ proved theoretically that, provided the effect of the Earth's magnetic field was neglected, the absorption of a radiowave of frequency f incident obliquely on the ionosphere at an angle of incidence i was equal to the absorption of a vertically incident wave of frequency f cos i multiplied by the factor cos i (Martyn's absorption theorem). Experimental investigations of the theorem were undertaken by Beynon² and Allcock³, both of whom concluded that its accuracy was considerably improved if the cos i multiplying factor was omitted. Later theoretical work⁴ supported the omission of the cos i term. The simple ray-theory on which Martyn's theorem is based breaks down at low frequencies because conditions in the ionosphere change appreciably within the distance of one wavelength. Allcock³ demonstrated experimentally, however, that to a first approximation the theorem was valid even at low (LF 30–300 kHz) and very low (VLF 10–30 kHz) frequencies. It has since become customary to express the reflexion coefficients of LF and VLF radiowaves propagated at oblique incidence in terms of vertical incidence propagation at an equivalent vertical frequency of f cos i and to assume that the reflexion coefficient will be the same for all paths having identical values of f cos i. Furthermore, effects resulting from changes of path direction relative to the Earth's magnetic field have also been neglected and no distinction drawn between the reflexion coefficient (∥R∥) and the conversion coefficient (∥R_⊥) for angles of incidence less than 50°.