462 2 PROPAGATION THROUGH THE STANDARD ATMOSPHERE 
while there is no radiation incident upon RF from the 
opaque surface CAPBD. In order to compute the 
field at R, it is assumed that in the open part of the 
plane the field is Hy while on the opaque screen the 
field vanishes. Such a field distribution may be 
realized physically by assuming that there is a con- 
ducting sheet in the open region HAPBF with 
suitably chosen oscillating charges or currents such 
that the field Eo is produced on the side of the sheet 
facing the receiver. The total radiation received 
at R from such a current-sheet will be equivalent 
to the radiation from 7 bent around the diffracting 
screen. The fictitious sheet HAPBF forms a system 
of secondary sources of radiation whose effect is 
equivalent to that of the primary source for all 
points on the far side of the plane ECDF (side of the 
receiver), but not on the near side (side of the 
transmitter). 
It is evident that most of the radiation received 
at R comes from the area near the point P above the 
line APB. The relative importance of contributions 
of areas more or less removed from P is discussed in 
following text. 
When the primary source at T is replaced by a 
distribution of secondary sources in the plane of the 
screen, an essential approximation is made. It is 
assumed that there are no secondary sources in the 
opaque region CAPBD. In reality, the screen is a 
physical body and, whether it is a conductor or a 
dielectric, there is an electromagnetic field in its 
surface layers, especially near the edge APB, and 
this field makes a contribution to the radiation 
received at R. In the approximate theory, it is 
assumed that the field on the surface of the opaque 
screen is negligible. In the terminology of optics, 
this implies that the screen is black; in radio termi- 
nology, it means that the surface of the screen is 
rough 
The Straight Edge Formula 
The physical picture just described can be put 
into mathematical language. When the rather 
intricate derivations are carried through, a rela- 
tively simple formula results. : 
The symbols and designations used are illustrated 
in Figure 2. In accordance with practice it is 
assumed that the line TR is nearly horizontal. The 
trace of the opaque screen on a vertical plane through 
T and R is assumed perpendicular to the line TR 
(upper part of Figure 2). The trace of the screen 
on a horizontal plane may, however, make an 
angle @ with the line TR (lower part of Figure 2). 
In view of the approximate nature of the theory 
explained in the preceding paragraph, the following 
conditions must be fulfilled in order to obtain 
reliable results: 
d,d2 > >ho >>. (1) 
ELEVATION 
PLAN VIEW 
\ 
\8 
Ficure 2. Diffraction around straight edge. 
That is, the distances from the transmitter and 
receiver to the obstacle must be large compared to 
the height of the latter above the line TR, and this 
height must be large compared to the wavelength. 
The second of these conditions is likely to be ful- 
filled in the short-wave and microwave bands, and 
the first will be fulfilled when the angles of elevation 
a and a, of the rays, drawn from the transmitter 
and receiver to the edge, are small. 
A second condition for the validity of the diffrac- 
tion formula refers to the horizontal extension of the 
screen. The formulas are derived for a screen of 
infinite horizontal extent, but in practice it will 
usually suffice if the horizontal extension of the 
screen is large compared to the height ho. 
If these conditions are fulfilled, the field at the 
receiver is given by 
jm /4 
V2 
where Ep is the asin field at the receiver in 
absence of the screen and 
p= thy? (142) 1.) = (asta). 3) 
In the last formula, use is made of the fact that 
ay ene a, are small angles so that approximately 
ho/d; and a2 = ho/ dz. 
é 
i = b= gee 2 du, (2) 
The Fresnel Integrals 
An integral of the type appearing in equation (2) 
is known as Fresnel’s integral; its properties will 
now be briefly discussed and numerical data given. 
The standard Fresnel integral is usually defined as 
C(x) — jS() = fe ee, (4) 
C(v) = [eos (< “) dv, 
so fan (ge)o 
If this function is plotted in the complex plane, 
with C and S as abscissa and ordinate, respectively, 
for all values of v, a curve is obtained that is known 
as Cornu’s spiral (Figure 3). C — jS is represented, 
where 
