This chapter presents a development of the theory of propagation of very short electromagnetic waves in free space, over a plane earth, and over a spherical earth, including the effects of the electromagnetic properties of the earth and refraction by the atmosphere. In a sense it is two chapters, for it presents detailed specific results and methods for computation of field strength in the absence of refraction or with a linear modified index profile, as well as theory and a few numerical examples for certain types of nonlinear profiles. In order to prevent duplication and to preserve generality as much as possible, however, the presentation has been designed to adhere closely to fundamental electromagnetic theory, from which specific results are developed as special cases. Because of the mathematical complexity of even the simplest cases of nonstandard refraction of practical interest the numerical results for these cases are few.

This chapter summarizes the principal experiments that have been performed to investigate the effects of atmospheric refraction on microwave transmission. Many of these experiments (particularly the earlier ones) were highly varied in nature and were not performed or analyzed by methods now in general use. They often revealed transmission properties that now appear to fit into a broad general pattern, although at the time this unity was not apparent. Consequently we do not follow a chronological approach, but instead we have arranged the material in a way that seems most appropriate in the light of present information. The point of view adopted here is that the modified-index distribution is the property of the transmission medium that is of major interest and usefulness. The marked dependence of the transmission phenomena on meteorological conditions makes it imperative that radio and meteorological observations be made simultaneously and that the weather regime of the experimental location be considered. Because the radio results must be interpreted in terms of meteorological conditions, statistical studies of even large amounts of radio data have little general significance and will be treated only briefly in this chapter. Our main concern will be the qualitative aspects of transmission as influenced by tropospheric refraction.

In the earlier parts of this book it has been assumed that radar targets could be characterized by the radar cross section σ, which has been treated primarily from the phenomenological point of view. In this chapter we shall examine the properties of σ and its relation to the properties of the target. It will be found that in only a negligible number of extremely simple cases is it feasible to calculate σ from the geometry; the remaining cases, which for the most part are those of greatest practical interest, are beyond the scope of existing mathematical methods.

In studying the propagation of microwaves through the atmosphere one is interested both in the absorption and in the dispersion, that is, how the index of refraction varies with frequency. A very general relation exists between the refractive index and the absorption coefficient that enables one to determine the absorption if the dependence of the refractive index on the frequency is known throughout the spectrum from v = 0 to v = °°. Vice versa, the refraction can be computed (apart from an additive constant O if the absorption is known for all wavelengths. In other words, assumptions cannot be made regarding how the refractive index varies with frequency without implications as to the amount of absorption. This fact has a bearing on the theory of nonstandard refraction. For instance, various investigators have noted that a variation of 1 per cent in the refractive index between λ = 1 and λ = 10 cm would lead to interesting differences in the trapping or duct phenomena at the two wavelengths. Such a variation, however, would necessarily lead to an unreasonably high absorption, in contradiction with experiment.

This appendix derives the upper bound of the ratio of the coherent to incoherent scattering. In comparison with a trapezoidal pulse of approximately the same “sharpness” it is seen that the smooth pulse has a much smaller coherent scattering.