CIIVKVCTKKISTIC \ M.i ES FOB THE BILINEAB M CI RVE 



231 



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 F(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 Ufa) U(zi) 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 

 in ^ except at great distances. Some other mode, 

 which does not suffer from the depth-loss effect, 

 although it may have greater attenuation, will be 

 the important mode at smaller distances. 



The conclusions for positive s cannot be expected 

 to apply unless the lower part of the M curve is 

 really sensibly straight over a considerable part of 

 its length. For negative s (trapping) this requirement 

 is not so important. 



It was mentioned that other models had been 

 employed by various investigators in calculating 

 field strength in the presence of a duct. The British 

 used an index distribution given essentially by Y = 

 (z — z m /m), where m lies between zero and unity. 

 When m = y 2 the problem could be treated by a 

 phase integral method, which Booker had done. The 

 differential analyzers at Manchester and Cambridge 

 had been used to obtain the characteristic values for 

 other values of m. The linear variation of index had 

 been studied by Hartree and Pearcey. In this case 

 of linear exponential variation Y = z + Ae~Bi, where 

 A and B are adjustable parameters. This model offers 

 the advantage that the index is an analytic function 

 of z and also that the modification term approaches 

 zero with increasing height. 



An alternative method (Langer's) for joining the 

 two parts of an otherwise bilinear M profile was 

 brought up. This method gives a solution in terms 

 of Bessel functions and solves the difficulty perfectly 

 for joining two straight lines. 



It was inquired whether, in case of positive s it 

 had been ascertained that for large g there were no 

 roots of the secular equation corresponding to a 

 linear M curve having the slope of the lower segment. 

 There was the possibility that the root found might 

 be one of a possible pair and that there might be 

 another solution of the wave equation for positive s 

 which had not yet been discovered. 



The author replied that the roots varied continu- 

 ously as g varied and that the investigation had 

 dealt with the root obtained when starting with the 

 first standard value for g = 0. What happened with 

 increasing g when the start was made from some 

 other standard value of g = was not known defi- 

 nitely, but the effects were believed to be peculiar. 

 It is expected that there may be some values lying 

 fairly near the s squared value for the imaginary 

 part. They are not considered to lie close to the s 

 squared value for the real part, as they would for 



