Chapter 20 

 INCIPIENT LEAKAGE IN A SURFACE DUCT 



201 CALCULATIONS FOR THE FIRST 

 MODE OF THE BILINEAR MODEL 



A recent interchange of ideas on problems of 

 mutual interest with members of the wave 

 propagation group of the Radiation Laboratory 

 prompted the author to investigate the variation of 

 the attenuation constant (or space decrement) a(h) 

 of the first mode with the duct height h and negative 

 index gradient a of a surface duct (see Figure 1). 



o 



2.0 



1.0 

 0.7 

 0.5 



0.3 

 0.2 



0.1 

 0.07 

 0.05 



0.03 - 

 0.02 - 



0.01 - 



0.001 



I 1 I I I I I I I I I I I 



S<1 



'z=o 



T 

 ft h»DUCT 

 _±_HEIGHT 



»Y(X) REFRACTIVE 

 INDEX ANOMALY 



0.007 - A F|RST APPR0X 



0.005 - 



_ O SECOND APPROX 



0.003 

 0.002 - 



I I I I I I l\ I 



* \ % 



\a=b \a=i/2b\a=i/4b - 



I i \l 



2 3 4 5 



Figure 1. Variation of the attenuation constant with 

 duct height. 



The attenuation constant is defined as the constant 

 a which occurs in the factor as: (l/-\/d) e~ ad , 

 giving the variation of the amplitude with range d. 

 The results are shown in Figure 1. In this figure 

 the attenuation constant a(h) is expressed in terms 

 of the attenuation constant for zero duct height 



"By C. L. Pekeris, Columbia University Wave Propagation 

 Group. 



a(0). In Figure 1 the curve for 5 < 1 was computed 

 from a formula developed by Freehafer and Furry 

 of the Radiation Laboratory: 



«(0) 



= 1 



C A A 6 



b 90 



1 + 



bj 1,260 



(1) 



where 



Here 



S = ft(fc 2 &) 4 . 



h = duct height; 

 a = —dM/dz inside the duct; 

 b = dM/dz above the duct; 

 k = 2x/X. 



It was felt that this equation could be used up to 

 5 equal to about 1.3 but not beyond this value. 



The curves on the right for 5 > 2, for which a 

 condition of nearly complete trapping is approached, 

 were obtained as follows. The secular equation for 

 the proper value of A (a ~ I m A), is 



g?(p) gf (8)//-f(g) + Hjf(s)H<f(q) 

 H ( |»(p) Hf(8)HJ$(q) + HJf(s)H^(q) 



0, (2) 



where 



q = 3a A '' P = 3a {A + ahy '> S = yq > y = b 



(3) 



is transformed by the substitution 



q = e wn x ,-p=: (at -|- /3)*, /3 = ah 



(4) 



into 

 with 



f(p) - F(x) = , 



Kp) = i. 



F(x) 



Hf(p) 



U(yx) V(x) + V(yx) U(x) 

 ~ e iT <* V(yx) U(x) - U(yx) V(x) 



U(x) = 7j(x) + e-'*l 3 I^(x) , 

 V(x) = I,(x) + e- /3 7_ ? (x) . 



(6) 



(7) 



233 



