170 TECHNICAL SURVEY 
show essentially exponential behavior in any region 
where the “height of the barrier,’’ i.e., the amount 
by which the Y curve lies to the left of the line at 
—B, is greater than A. 
This sort of general physical consideration about 
the U curve leads, on being put in more precise 
mathematical form, to the phase integral methods. 
Unfortunately, no phase integral method can claim 
validity for this model except in cases of trapping. 
In general, the Eckersley phase integral method for 
untrapped modes requires that the Y curve be an 
analytic function, and the bilinear curve obviously 
is not. Most of the values presented are, accordingly, 
the results of exact calculation. 
Figure 3 shows that for negative s the attenuation 
falls rather suddenly to very small values at a certain 
value of g and then quickly approaches zero. This 
indicates the occurrence of trapping. On the other 
hand, for positive s, the attenuation constant 
approaches a finite asymptotic value. It is interesting 
to note that this is always definitely less than the 
value s’x standard, which corresponds to a single 
straight line of slope s?x:standard slope. 
It is also useful to know the real part B of the 
characteristic value. Figure 4 shows the complex D 
plane. For g = 0 the Y curve is just standard, and 
as g increases the value of D for each value of s 
traces out a curve; for small values of g all these 
curves practically coincide. For negative s the real 
part decreases steadily as soon as the imaginary 
part becomes very small. For positive s, on the other 
hand, the real part as well as the imaginary approaches 
a finite limiting value, so that each curve has an 
end point. 
Some of the consequences of this behavior of the 
real part can be seen by studying Figure 5. The first 
row of diagrams shows the situation for fixed nega- 
tive s and increasing value of g. The first diagram 
shows the standard curve. The next shows a curve 
with a small superstandard section, but — B still lies 
in nearly the same location relative to the dotted 
line which marks where the origin lay for the stand 
ard curve; thus B has increased. The first. figure o 
Wek BS 
“BO 
Ficure 5. Curves for negative and positive s. 
the second line shows how the same thing happens 
for a small substandard section. Thus for small g 
the first order effect is just to add the amount g to 
D, for all values of s. 
In the third diagram of the first row we have a 
case in which the superstandard has a pronounced 
effect, but trapping has not yet set in. In such inter- 
mediate cases B may become positive, but the 
diagram shows a case in which it happens to be 
zero. In the fourth diagram trapping is definitely 
established; B has become negative, and the line 
—B has taken on a definite position relative to Y (0) 
(dotted line). This same relation is maintained for 
larger values of g, as in the last diagram of the top 
row. In the last diagram the “barrier” has become 
much more formidable. This means that the value 
of U just above the barrier is extremely small, and 
thus the attenuation is very small because of the 
small leakage. 
In the second row, as has been remarked, the first 
diagram shows a small substandard section which 
has only a small perturbing effect; — B lies essentially 
at the standard distance from the intercept of the 
extrapolated standard curve. The second shows an 
intermediate case. In the third diagram the limiting 
value of D has been reached, and the line at —B 
has taken its final position relative to the joint of 
the Y curve. In the last, larger, diagram g has 
become much larger, but —B has still the same 
position relative to the joint. 
The difference between the last two diagrams is 
essentially the increase in the strength of the surface 
barrier. The structure of the height-gain curves near 
and above the joint is practically the same in the 
two cases. The very thick barrier in the last case 
causes the intensity near the earth’s surface to be 
extremely small. This particular kind of height-gain 
effect can be more suggestively referred to as depth 
loss. The amount of this depth loss is very large: 
the first 200 ft of the substandard layer produces a 
loss in the product U(2:) U(ze) of at least 200 db (at 
10 cm), which is about four times the gain for a 
similar height in the standard case. Moreover, this 
loss proceeds at a rapidly accelerating rate, whereas 
standard height gain goes at a decreasing rate. The 
same situation of depth loss in thick nonstandard 
layers occurs in transitional cases, with s positive 
but less than unity. 
In general the results for the first mode for positive 
s can be summarized as follows: 
In nonstandard layers of fairly small thickness, 
less than 100 ft for 10-cm waves, the propagation is 
not markedly different from standard for the sub- 
standard case and can have attenuation strikingly 
less than standard for suitable thickness of a transi- 
tional layer. 
For thick layers there is a strong depth-loss effect 
in the first mode in both sorts of cases, and the first 
mode cannot be expected to be the dominant term 
