62 TROPOSPHERIC REFRACTION 



The integral for r, (3.2), cannot be evaluated directly without a knowl- 

 edge of the behavior of n as a function of height. Consequently, the 

 approach of the many workers in this field has been along two distinct 

 lines: the use of numerical integration technicjues and approximation 

 methods to evaluate r without full knowledge of 7i as a function of height, 

 and the construction of mode) ?i-atmospheres in order to evaluate average 

 atmospheric refraction. The following sections are devoted to a discus- 

 sion of these methods. 



3.2. Limitations to Radio Ray Tracing 



The user should keep in mind that the equations given in the preceding 

 section are subject to the following restrictions of ray tracing: 



(1) The refractive index should not change appreciably in a wavelength. 



(2) The fractional change in the spacing between neighboring rays 

 (initially parallel) must be small in a wavelength. 



Condition (1) will be violated if there is a discontinuity in the refractive 

 index (which will not occur in nature), or if the gradient of refractive 

 index, dn/dr, is very large, in which case condition (2) will also be violated. 

 Condition (1) should be satisfied if 



(dn /dh) per km 



T^ < (J.VVZjkc, 



where refract ivity, A^, is defined as .V = (n — 1) X 10^ and/kc is the carrier 

 frequency in kilocycles per second [2]. Condition (2) is a basic require- 

 ment resulting from Fermat's principle for geometrical optics. An atmos- 

 pheric condition for which both conditions (1) and (2) are violated is 

 known as "trapping" of a ray, and it can occur whenever a layer exists 

 with a vertical decrease of A'' greater than 157 A'^-units per kilometer. A 

 layer of this type is called a "duct," and the mode of propagation through 

 such a layer is similar to that of a waveguide [3]. Taking into account 

 refractive index gradients, a cutoff frequency may be derived for wave- 

 guidelike propagation through a ducting layer [4]. 



In addition to the above limitations, it should be remembered that the 

 postulate of horizontal homogeneity, made in order to use (3.1), is not 

 realized under actual atmospheric conditions; some degree of horizontal 

 inhomogeneity is always ])resent (see chapter 8). 



