THE SOLUTION OF THE PROPAGATION EQUATION 



239 



I. There is an infinite number of these "Eckersley" 

 or "leaky" modes. Equation II joins on smoothly to 



-1.0 



-0.5 



0.5 



Figure 1. Diagram showing the locus of p in the com- 

 plex plane of a varies. 



I and defines a finite number of "transitional" or 

 "semileaky" modes. 



Equation III defines the strongly trapped or 

 Gamow modes for which < I m (p) <K — Re(p) < 

 1. In this case the characteristic values lie almost 



upon a Stokes line (arg p = t — 8). The approxi- 

 mations valid above the line yield II; the ones valid 

 below yield 



IV 1 + ie-w + »' = , or 



( P + l)» = (n - %) -< 1 (n = 0,1,2- • •)• 



Thus in the Gamow case IV makes I m (p) = while 

 II yields almost the same real part, but a very small 

 positive imaginary part for p. The mean approxima- 

 tions effectively yield III whose roots are essentially 

 the mean of the roots of II and IV. This averaging 

 process checks closely with an exact calculation based 

 on the (unpublished) WPA tables of Bessel functions 

 of order ^ for real and pure imaginary arguments. 

 It is to be noted also that IV determines the number 

 of strongly trapped modes as the largest integer n 

 such that (n — l A) ir/cr < 1. 



The characteristic value problem may be regarded 

 as essentially solved in the cases of leaky modes (I) 

 and strongly trapped modes (III). In doubt still is 

 the question of the transition between III and II; 

 this uncertainty plus the fact that actual determina- 

 tion of the roots of II is much more complicated 

 than in the other cases, and that in case II the 

 arguments of certain of the Hankel functions lie so 

 close to the origin as to make doubtful the validity 

 of the asymptotic procedure — all these considera- 

 tions indicate that resolution of the transitional mode 

 question awaits appearance of adequate tables of 

 the Hankel functions for complex arguments. 



