linear theory must be based on a well-developed linear theory, including the amplitude 

 distribution. 



While the difficulty has been generally recognized by workers in this field, Toll- 

 mien [19] was the first to attempt a solution to the problem. His solution is restricted 

 in three ways: (a) only the case of neutral oscillations is treated; (b) only the first 

 approximation is given and there is no obvious algorism to obtain the higher approxima- 

 tions; (c) only the function and its first derivative are considered,. Wasow [20] treated 

 the general complex case including damped and amplified oscillations, but his treatment 

 was not free from the restrictions (b) and (c). There is also an obvious difficulty 

 in reconciling his "'uniformly valid solution" with his own solution away from the 

 singular point. This difficulty can, however, be easily removed, e.g., by using the 

 transformation [15] used in the method to be described. I shall not go into further 

 discussions of these issues, because Dr. Wasow is here to present a comparison of 

 the various solutions including another one due to Langer. I wish merely to state 

 that the solution to be presented appears to be simple, complete, and most readily 

 applicable. 



a. Older methods 



Let us begin by considering the older methods. For aR large, it is known that 

 formal asymptotic solutions of (4) may be found as follows: 



1 

 = 0<»(y) + — *»%) + • ■ ■ , (8)' 



aR 



+ = e ± vSg (l ,) f( o )(lj) ± ——fW(y) + • ■ • I (9) 



VaR J 



where 



Q(y) = j y \i(w - c)]* dy, pKy) = (« - c)-^, • • • (10) 



and y c is the critical point where w(y) ~ c. One of the formal asymptotic solutions of 

 the form (8) (commonly denoted by ^J is regular, while another (commonly denoted 

 by </>,) exhibits a logarithmic behaviour at the critical point y — y c . The singularity in 

 this asymptotic solution and the solutions of the form (9) shows that they can be valid 

 only when the immediate neighborhood of the critical point y c is excluded. The multiple- 

 valued nature of the singularity shows further that the region of validity must be 

 restricted to certain sectors in the complex plane in the neighborhood of the point y c . 

 One has therefore to study very carefully the nature of the solution in that neighbor- 

 hood. In the older theory, this is accomplished by a change of scale, i.e., by the intro- 

 duction of the new variable 



r, = (y - y c )/e, (11) 



followed by expansion of the solution of (4) in the form 



4>(y) = x(n) = x (0) W + ex (1) W + • • • (12) 



Such expansions are valid for finite values of 77. It has been shown, [11] however, that, 

 for large 77, the asymptotic expansion of (12) can be formally identified with the solu- 

 tions (8) and (9) after regrouping of terms. 



Two things become clear from such an investigation. First, to obtain <j> i0) (y), 

 we need all the terms in (12) showing clearly that a few terms from (12) will not be 

 accurate enough for y — y c finite. Similarly, a few terms from (8) will not yield accurate 

 results for finite values of 77, i.e.. for y — y c — O(e). Unfortunately, for many applica- 



358 



