DIFFRACTION BY TERRAIN 463 
in magnitude and phase, by a vector from the origin 
to a point on this spiral. 
It may be shown that the length of are along the 
spiral, measured from the origin, is equal to »v. 
In the graph, values of v, counted positive in the first 
quadrant and negative in the third quadrant, are 
indicated along the spiral. As v approaches infinity, 
the spiral winds an infinity of times around two 
points lying at the distance 1/¥2 from the origin 
on a 45-degree line. C and S for the end points are 
Cle) =>, Sea aee 
Application to Straight Edge 
Since 
et ltj 
Somme 
equation (2) may be rewritten, on using equations (4) 
and (5), as 
B=% #1 h 4 cw -L-jsm|. © 
It will be noticed that the quantities 
1 -f1 
=~+C, j(=+8S 
pied ¢ t ) 
have a simple geometrical meaning. They are the 
real and imaginary components, respectively, of a 
Figure 3. Cornu spiral. 
vector drawn from the lower point of convergence 
(point —1/2, —j/2) of the Cornu spiral to a point 
on this spiral. The bracket in equation (6) is equal 
to this vector in magnitude but with opposite phase. 
The Fresnel formulas and Cornu spiral as given 
above will assist the reader in establishing the rela- 
tions of our equations with the classical theory of 
diffraction as found in all textbooks on the subject. 
For practical purposes the field behind a diffracting 
straight edge given by equation (6) will be denoted by 
E = 14 . 
Ei ze"?, (7) 
In Figure 4, fhe modulus z is plotted as a function of 
v. In Figure 5, the phase lag ¢ is plotted in a similar 
way. (With the above choice of the sign, ¢ is positive 
in the shadow.) 
The variable » is given by equation (3). On 
account of the square root, there is an ambiguity in 
sign. Closer inspection shows that » must be taken 
positive when the receiver is in the illuminated region, 
above the line of sight; » must be taken negative when 
the receiver is in the shadow Zone. 
When »v tends to —  , the line APB (Figure 1) 
moves far upwards relative to the line TR; the 
receiver lies deep in the shadow and # approaches 
zero by equation (6). When» tendsto + o , the line 
APB moves far downward, and the screen ceases 
to form an obstruction, H approaches EH. At the line 
of sight (when the point P in Figure 1 coincides 
