SECOND AND 1 1 1 « ; 1 1 10 It MODES OF THE BILINEAR MODEL 



235 



too but beyond g = 2 is seen to level off to a constant 

 limiting value. The real part of the characteristic 

 value B« also approaches a constant limiting value 

 for g greater than 3. These curves were obtained by 

 solving the secular equation for D and also deter- 

 mining the slope dD/dg at each point. The charac- 

 teristic values curves can also be computed by 

 starting first with Gamow's values appropriate for 



Figure 3. Characteristic values D m for a bilinear M 

 curve, s = — V2. g = height of joint in natural units. 

 s 3 = ratio of slope of lower segment of M curve to the 

 standard slope. D m = B m + i A m . 



large g and continuing backwards toward smaller 

 values of g, being careful to determine the slope of 

 the curves at each point. It is seen from Figures 2 

 and 3 that, in contrast to the first mode, these 

 branches of the curves for the second and . third 

 modes do not join on smoothly to the other branches 

 which start with standard values at g = and 

 approach limiting values for large g. 



This duplicity of the solutions, which was doubted 

 at first, was substantiated in two ways. The values 

 of Ai and B 2 at g = 3 and g = 5 in Figure 1 were 

 computed at both branches with increasing accuracy 

 (up to 10 -6 ), and it was found that the matching of 

 the solutions at the duct height and the degree of 

 vanishing of the height-gain function of the ground 

 improved correspondingly in both branches. This 



Table 1. Comparison of exact limiting values of D with 

 values obtained from the asymptotic formula.* 



Second mode 



Third mode 



-1 



V'2 



-0.60 + 2.80i 

 -0.59 + 2.83i 

 -0.78 + 2.74i 

 -0.70 + 2.70i 

 -1.00 + 2.60« 

 -0.80 + 2.44i 



-1.06 + 3.60i 



-1.22 + 3.40i 

 -1.42 + 4.08i 

 -1.36 + 3.48i 



Asymptotic 



Exact 



Asymptotic 



Exact 



Asymptotic 



Exact 



*exp 



-?»' 



1 - s 3 /%D'i = . 



proves that both solutions satisfy the boundary 

 conditions. As a second step in testing the reality of 

 the limiting points, an asymptotic expression was 

 derived for the limiting values, and the values com- 



IOOO 



too 



10 



t i 



0.1 



O.OI 



3 4 



Figure 4. Height-gain functions of the second mode 

 for a bilinear M curve, s = — V 2. z = height above 

 ground in natural units, g = height of joint in natural 

 units. Ut (z) = normalized wave function. 



