Scveral numerical examples are presented to show thc prcdominant features of radio wave propagation over an inhomogeneo us earth . These are based on the tllCories d eriv ed previously [Furutsu, 1957a[Furutsu, , 1957b[Furutsu, , 1959[Furutsu, , 1963 in which the h eight and also the elect rical properties of the earth's surface were assumed to change disco ntinuously several times along the wave path; thus, t he terrain represented could include ridges, cliffs, blufJ"s at a coastli ne, ete. Th e theory is briefly reviewed, and num erical res ults are prese nted for the spherical earth approximation and are compared to those for t he flat eart ll approximation. For a p e rfectly conducting flat earth , there a re well-known formu las available in t erms of t he Fresnel integral, and the sphe rical earth results are compared to those obtain ed using these formulas to show the agreeme nt at short distances. A few interesting phenom e na are also illustrated, such as the obstacle gain d uc to a ridge on a lossy g round and the variation of fi eld strength causcd by a change of receiver (or transmitter) height when the wa ve is propagated over a mixed path. Finally, set s of graphs are a lso in clu dC'cl to aid in evaluating the effects of a ridge or a cliff on a homogeneous earth ; they can be used when the propagation distances are suffic iently large 011 each sid e of t he rid ge or clifr.
The problem of determining path a nte nna gain is t reated here in greater detail t han pre viously [1 , 2, 3).1 The m ethod used here takes in to acco un t for t he first t ime t he ex ponent ial decrease of the gradient of refractive index with height , a nd a scatterin g cr oss section inversely proportional to t he fifth power of t h e scatterin g a ngle. R es ults are given for all combinations of beamwid ths fi nd p ath geom etry, assuming t hat symmetrical beams are used on both ends of t he path a nd t hat atmospheric t urbulen ce is isotropic. The result appears as a function of both of t he beamwid ths, in addition to othe r para meters, a nd thus t he loss in gain cannot be determin ed i ndependently for the t ransmi tting fi nd receiving antenn as. The va lues of t he los in gain are generally lower t han t he previ ous estimates for which a comparison is poss ible.
Several approximations to the attenuation coefficient caused by double knife‐edge diffraction are given in terms of the single knife‐edge attenuation coefficient. These approximations are evaluated for several typical double knife‐edge paths and compared with the theoretical value in order to determine the best approximation in the different regions of interest.
When radio rays pass th rough t he atmosphere, they are defoc Llsed due to its presence.This effect is measu red by the divergence coefficient a nd general formulas are derived for D1, the dive rge nce coefficient of t he direct ray, and D" the divergence coefficient of the reflected ray-assuming a smooth spherical earth . As examples, D I a nd Dz are shown for some typical cases with an "exponential" atmosphere (troposphere).
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