138 TECHNICAL SURVEY 
space values. The angles at which the minima occur 
lie half way between. The scale of vertical distances 
is greatly exaggerated compared with the horizontal 
scale. Coverage diagrams for the same frequency and 
transmitter height, but taking account of the earth’s 
curvature, are shown in Figure 24. 
Coverage diagrams for more complicated situa- 
tions must take into account, in addition to the 
factors already mentioned, the curvature of the 
earth, the refraction of the atmosphere, and diffrac 
tion into the region below the line of sight. 
Ficure 6. Use of equivalent ground plane. 
When the ground is sloping, the above construction 
may be modified as indicated in Figure 6. For any 
specified lobe, determine approximately the part of 
the ground where reflection takes place. Draw a 
tangent to the ground in this region and determine 
the perpendicular projection of the antenna site or 
this plane (“equivalent ground”’). Use the equivalent, 
height thus determined in equation (8), and let the 
angle 6 refer to the plane of the equivalent ground. 
This procedure is also required when the transmitter 
and receiver or target are of comparable height so 
that the reflection point is not near the transmitter, 
When the transmitter is set up near a coast, the 
lobe pattern over the ocean will undergo periodic 
variations caused by the tides. Since, in equation (8), 
B is multiplied by hi, it follows that the lobes will be 
low at high tide and high at low tide. This phenom- 
enon may become very important for heightfinding 
sets. 
A more complicated case occurs if ground reflec- 
tion is not complete. Then p is less than| unity, and ¢ 
differs from 480°. In this event*the lobes have max- 
FREE SPACE 
FIELD 
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rr) FLAT EARTH 
Fieure 7. Coverage diagram for incomplete reflection. 
ima which are less than twice the freé space field and 
minima which never reach zero. The angular posi- 
tions of the lobes are changed somewhat, but the 
most noticeable change is found on the lower side of 
the first lobe. It is likely to lie at a lower elevation 
and reaches the ground at some distance from the 
transmitter (compare Figures 5 and 7). 
Refraction—Snell’s Law 
The bending of rays in the atmosphere depends 
upon the refractive index n which is a function of the 
temperature, pressure, and moisture content of the 
air. The manner in which these quantities control the 
index of refraction is explained on pages 142-143 Toa 
first approximation, assuming horizontal stratifica- 
tion of the atmosphere, the index may be considered 
to be a function only of height above the ground. The 
corresponding case, familiar in optics, is that of two 
media, such as water and air, with different refrac- 
tive indices m and m_(Figure 8A). If a1 and a2 are the 
angles between the rays and the plane of the bound- 
ary, Snell’s law of refraction states that 
N, COS a1 = Ne COS Ao. 
In the atmosphere the refractive index changes 
continuously with height. The simplest case, often 
2 
Oy BOUNDARY REFERENCE. 
n=n(h) v1 
A 8B 
Fieure 8. A. Refraction at a sharp boundary. B. Re- 
fraction through a layer with variable n. 
encountered in practice, is that of a refractive index 
which decreases linearly with height. This is known 
as standard refractions Snell’s law applies here also, 
since the atmosphere may be divided up into an in- 
finity of parallel boundaries, the change of refractive 
index from one boundary to the next being infinites- 
imally small. Instead of a sudden change of direction 
there is then a gradual change or bending of the rays 
(Figure 8B). Snell’s law may then be stated gen- 
erally as 
nN COS a= COS ao , 
where now 7 and @ are continuous functions of height 
and the zero subscript on the right-hand side refers to 
any fixed reference level. The curvature of the re- 
fracted rays is downwards or upwards according to 
whether the refractive index decreases or increases 
with height. 
Refraction over a Curved Earth 
In reality the surfaces of constant refractive index 
are not planes but are contentric spheres about the 
earth’s center. In this case Snell’s law assumes a 
