234 



INCIPIENT LEAKAGE IN A SURFACE DUCT 



Assume now that 



P = Po 



(8) 



where p is a constant, which is to be chosen in such 

 a manner that A is small in comparison to p in the 

 region under consideration. Expanding equation (5) 

 in a power series in A, one obtains as a first approxi- 

 mation for A : 



Ai = 



F(x ) - /(p„) 

 /(Po) 



(9) 



and for a second approximation 



A 2 = Ai 



1 - 



+ 



F(x B ) ^ 

 dp 



Ai/(p ) 



2/(po) ' /(po) 



]■ 



(10) 



These expressions can be computed with the aid of 

 the WPA Tables (unpublished) for the I functions 

 with real argument. The curves in Figure 1 were 

 computed down to values of 5 such that A 2 did not 

 deviate appreciably from Ai. 



Conclusion. From the computed attenuation for a 

 surface duct it appears that, for the first mode, 

 when 5( = hkV) is less than 1, trapping is less than 

 2 per cent and that when 5 = 3 to 5 (depending on 

 the negative gradient a), trapping is 98 per cent 

 complete. (For the meaning of the constants see 

 Figure 1.) There is therefore a rather narrow range 

 of values of the parameter 8 (1 to 4) within which a 

 rapid transition takes place from a condition of 

 negligible trapping to a condition of nearly complete 

 trapping. This result may have a bearing on the 

 observed fading which is associated with ducts. 



202 CALCULATIONS FOR THE SECOND 

 AND HIGHER MODES 

 OF THE BILINEAR MODEL 



The Analysis Section of Columbia University Wave 

 Propagation Group has undertaken the computation 

 of the characteristic values and height-gain functions 

 for the second and higher modes of a bilinear model 

 M curve. The first mode of the bilinear model is being 

 treated at the Radiation Laboratory. The computa- 

 tions were carried out with the aid of tables of h 

 functions prepared by the Harvard Computation 

 Laboratory, under the direction of Furry. Our work 

 to date has been mainly on surface ducts, in which 

 the slope of the lower segment of the M curve is 

 negative. Cases with positive slopes of the lower 



segment of the M curve have been tried but were 

 found to involve h functions which are beyond the 

 range of existing tables. 



Some results on the characteristic values are shown 

 in Figures 2 and 3 (for a definition of natural units 

 see preceding articles). In Figure 2 the slope of the 

 lower segment of the M curve is the negative of the 

 standard slope, while in Figure 3 the ratio of the 

 slope of the lower segment to the standard slope is 



-2 



-3 



E 

 4 



Figubb 2. Characteristic values of D m for a bilinear 

 model, s = — 1. D m = B m + i A m . s 1 = ratio of slope 

 of lower segment to standard slope = s 3 . g = height of 

 joint in natural units. 



— \/8. The curves Ai and Bx for the first mode were 

 computed at the Radiation Laboratory. An imagi- 

 nary part Ai, which is proportional to the horizontal 

 attenuation (decrement), starts at g = with a 

 value appropriate for a standard atmosphere and 

 decreases continuously as duct height g increases. 

 Beyond g = 3 the first mode is completely trapped. 

 The curve A 2 for the second mode decreases initially 



