88 TECHNICAL SURVEY 
C = —0.25 to C = +0.25. The resultant R = 0.5 
substituted in equation (37) gives a power intensity 
of % relative to the unobstructed wave and a field 
strength of 0.353. 
The field intensity at P’ is due to the same length 
Av but taken over a different portion of the spiral. 
For this purpose, it is desired to use distances along 
the plane PP’, x, instead of s (Figure 30). 
pe ee eo (44) 
a 2a 
Thus the portion of the spiral nO in Figure 31 from 
v = 0.9 tov = 1.4 has an average value of v = 1.15 
which multiplied by the radical term of equation (44) 
gives x. The chord connecting these points is 0.43, 
and the relative power intensity is 0.092. This same 
result may be computed from the table of Fresnel 
integrals by obtaining the values of AC and AS for 
v = 0.9 and » = 1.4. The sum of the squares of AC 
and AS is R®. Typical patterns for slits of several 
widths are shown in Figure 32. It will be noted that 
there is little radiation outside the slit. 
Diffraction by a Narrow Obstacle 
The effect of a narrow object with parallel sides 
may be determined with the Cornu spiral. In the 
ease of the slit only a fixed length slid along the 
spiral is effective, the remainder being shielded by 
the edges of the slit. With an obstacle, however, a 
fixed length slid along the spiral represents the 
ineffective portion. If the obstacle is of such size 
that it covers an interval Av = 0.5 on the spiral, 
Figure 31, the segment Av may be located as JK. 
The radiation at the point considered will be due 
to the two parts of the spiral Z’ to J and K to Z. 
The resultant amplitude is obtained by adding the 
two vectors Z’J and KZ. The sum is RF for a point 
Av=l5. bv? 4.6 
-5 O +5 -—3 © +3 
4v=2.5 Av=5.2 
-4 0 +4 -3 Oo +3 
4y=3.9 Av=6.2 
-6 -3 O +3 +6 
Ficure 32. Diffraction patterns of slits. 
-5 to) 
CE VV 
Ficure 33. Diffraction of narrow obstacles. 
midway between J and K. The head of the vector 
is always in the direction Z along the spiral. Typical 
patterns for narrow obstacles are shown in Figure 33. 
Multiple Slits and Obstacles 
Slits or obstacles with parallel sides may be treated 
by means of the Cornu spiral and the resultant sum 
of the vectors obtained. Thus, with two slits of a 
width such that Av = 0.5 and spaced so that Av =0.5 
may be located on the spiral as JK and lm in 
Figure 31. The total R is the vector sum of R and 
Rk”. The field strength pattern is then obtained by 
sliding the two lengths along the spiral holding their 
spacing fixed. 
In similar fashion two narrow obstacles would 
cause two absent sections such as JK and Im and 
three open sections Z’J, Kl, and mZ. The three 
vectors, obtained by joining these three latter pairs 
of points, are combined to give the resultant ampli- 
tude R. 
Limitations of Fresnel’s Theory 
Neither Huyghens’ principle nor Fresnel’s theory, 
on which the above treatment is based, is rigorous, 
and their limitations must be kept in mind when 
making applications to radio and radar problems. 
In the development of the theory no mention was 
made of the effect of the shape and composition of 
the edge. Actually within a region of about one 
wavelength around the edge the wavefront is 
affected by the presence of the edge. In Figure 34 
the region of the edge disturbance is DE, and first 
half period of the wave front is DF. The first half 
turn of the Cornu spiral is due to DF. The position 
of F depends on the point considered. When DE is 
an appreciable part of DF, the simple Fresnel theory 
should not be depended upon. This occurs when the 
