466 2 PROPAGATION THROUGH THE STANDARD ATMOSPHERE 
to the height of the latter above the straight line 
TR; and that height should be large compared to 
the wavelength. 
Moreover, as discussed in the first section , the 
diffraction formula applies in principle only to the 
case where the effect of the currents induced on the 
surface of the ridge upon the field at the receiver 
can be neglected. This is the case (1) when the ridge 
has the shape of a steep and narrow knife-edge 
protruding from the surrounding countryside; or 
(2) when the surface of the ridge is rough (see sec- 
tionbelow). Experience shows that so long as the 
profile of the ridge is reasonably compact and its 
surface reasonably rough, the diffraction formula 
will give the magnitude of the field behind the ridge 
to within a few decibels. 
If the ground near the transmitter or receiver is 
smooth, however, it becomes necessary to take 
ground reflection into account. This may be done 
by introducing an image transmitter and receiver. 
The field is then the sum of four components whose 
relative phase must be calculated (see Figure 11). 
Earth curvature will be neglected throughout the 
present section. 
Criterion for Roughness 
It is difficult to establish a quantitative criterion 
for the roughness of a surface. From the viewpoint 
of radiation theory, the effect of a rough surface is 
to scatter incident radiation diffusely in all directions 
with no preference for the direction of regular reflec- 
tion, whereas a smooth surface will reflect the inci- 
dent radiation according to Snell’s law. In radio 
work, the effect of diffuse reflection is to weaken the 
radiation scattered in the direction of the receiver 
so much that its intensity may be neglected com- 
pared to the direct ray. A moderately rough surface 
will give a coefficient of reflection intermediate 
between zero and unity. A surface will be optically 
smoother as the incident radiation approaches 
grazing, and even surfaces that are comparatively 
rough geometrically may then give partial reflection. 
A rule taken from optics and known there as 
Rayleigh’s criterion has been used successfully in 
radio practice. Assume that the roughness is pro- 
duced by numerous small elevations above a level 
surface and let H be the typical height of such an 
elevation. The difference in path between a ray 
reflected from the ground and a ray from the top 
of the elevation is 2AB in Figure 7, which is equal 
to 2H sin y or 2Hy approximately for small angles y. 
The difference in phase between the two rays is 
2Hy(27/X). The criterion now requires that the 
surface be considered as rough when this phase 
difference exceeds 45 degrees, or 7/4 radians. 
Hence the critical value of H is given by 
4nrHy) 7 r 
FU H=—, 
7 eae aT Ey (8) 
Ficure 7. Geometry for Rayleigh’s criterion for 
rough ground. 
with y in radians and 
t= @) 
with y in degrees. The surface is considered smooth 
or rough according to whether H is smaller or larger 
than this value. 
Sometimes it is convenient to refer to the field 
pattern that would be present over a reflecting 
surface. This is done by introducing a new variable, 
the lobe number 
Ay 
n 1 (10) 
(fd transmitter height above the ground), where 
n = 1, 8, 5, etc., correspond to the angle of the first, 
second, etc., maxima in the lobe pattern and n = 0, 
2, 4, etc., to the nulls of the lobe pattern. Introduc- 
ing 7 into equation (8), the criterion assumes the 
form 
ha 
eS (11) 
Although the criterion is approximate and gives no 
more than an order of magnitude estimate, it is rather 
surprisingly well fulfilled in radio practice. Experi- 
ence has shown that when the differences in level 
which constitute roughness are of the order indicated 
by these equations, the reflection coefficient is 
reduced to a small fraction (about one-fifth) of the 
value calculated for an ideal surface. 
H= 
Diffraction by a Straight Ridge 
Assume that the ground intervening between the 
transmitter and receiver is everywhere rough, so 
that all ground reflection may be neglected. For 
the sake of computation, the ridge is replaced by a 
vertical screen of height ho above the line TR. The 
top of the ridge forms the diffracting edge (P in 
Figure 8). If the profile of the ridge is somewhat 
‘more complicated, the effective diffracting edge might 
be a purely mathematical line, as shown in the lower 
Ficure 8. Diffraction by a straight ridge. 
