234 REFLECTION AND TRANSMISSION OF RADIO WAVES 



million. Numerically, 10^ hja amounts to 0.048 M unit per foot. From 

 Equation 4-82 



f = (:| + ^)x.O' (4-83) 



It follows from this that a homogeneous atmosphere {dn jdh = 0) has an 

 M curve with dM jdh equal to lOV'^, and that an atmosphere with a 

 constant gradient of refractive index is equivalent to a homogeneous 

 atmosphere of effective radius <3e, where 



- = - + % (4-84) 



a, a dh 



In temperate climates an average value oi dn jdh is about —\/{4a). Hence 

 from Equation 4-84 



ae = \a (4-85) 



which is the so-called "four-thirds earth." Such an atmosphere is known as 

 the standard at7nosphere, and the corresponding M curve, which is a straight 

 line of slope 0.036 M unit per foot, as the standard M curve. Actually the 

 M curve is rarely a straight line except in a restricted height range. 



The M curve is useful in ray tracing, since a one-to-one correspondence 

 exists between the change in slope of a ray over a height interval and the 

 change in M. In fact, if represents the elevation angle, measured in mils 

 (1 mil = 10~^ radian = 3.44 minutes of arc), at height /z where the modified 

 index has the value M, and 0o, Mo are the corresponding quantities at a 

 reference height h^ (such as the ground), then 



Q = V^o^ + 2(M- Mo). (4-86) 



It can be seen from this that a height interval over which M — Mo is 

 negative will give rise to a decrease in the absolute value of the elevation 

 angle. Also, if the M curve has a sufficiently large negative excursion 

 (Mo — Mmin > ^0^/2), then the ray will become horizontal at a certain 

 height, and then curve back to earth. Assuming no loss in reflection at the 

 earth's surface, the process will be repeated over and over, and the ray will 

 go through a succession of hops along the surface. The ray is then trapped 

 between the earth's surface and the height at which it becomes horizontal. 

 A region of the atmosphere within which certain rays are trapped is called 

 an atmospheric duct. The multi-hop trajectory resembles somewhat the 

 crisscross path between the walls in waveguide propagation; and like a 

 waveguide, an atmospheric duct can trap only waves of frequency higher 

 than a lower limit. For effective utilization of the duct, both the radar and 

 the target should be within the duct. 



