tions, (including the case of the boundary layer) one boundary point corresponds to 

 finite or large y — y c while the other corresponds to finite -q. Thus neither (8), (9) nor 

 (12) can be adequately used for the solution of a characteristic value problem. This is 

 a central mathematical difficulty in the stability theory. 



The second point is that the asymptotic matching is restricted to certain sectors 

 of the complex plane. This leads to the conclusion (later confirmed by more careful 

 mathematical investigations) that the damped and the self-excited solutions are not 

 complex conjugates in the limiting case of infinite Reynolds number even though the 

 invicid equations appear to lead to that conclusion. 



We shall not go into further details of solutions of the type (12) as it will not 

 be used in the following investigations. 



b. Uniformly valid asymptotic solutions of the Orr-Sommerfeld equation. 



To obtain uniformly valid asymptotic solutions of the Orr-Sommerfeld equation, 

 we apply a theory developed for a class of differential equations of that type. In fact, 

 we need only to know the structure of the solution to obtain the desired results. This 

 will now be presented. The general theory will be presented elsewhere. 



According to the general theory, asymptotic solutions of the Orr-Sommerfeld 

 equation may be found in the form 



4> = K Q u + Km' + \-'\K 2 u" -f K 3 u'"), (13) 



In this formula, u is a solution of the equation 



u™ + Y-{zu" + yu' + pu) = (14) 



7 = X-27 ( i } + • • • (14a) 



= /3< 01 + X-2/3<i> + • • • 

 where z is related to y by 



z = \%J[i(w-c)]idy]*, (15) 



y 

 c 



and the parameters A 2 and (3 {0> are given by 



A- = <xR, /3<°> = - (w c "/w c ')(iw c ')-i (16) 



In (13) we also have 



Ki = KiW + X- 2 2^<« + • • ■ i= 1,2,3,4 (17) 



where Kj (j) (z) are regular in z. We note that to each solution u(z), there is a corre- 

 sponding solution 0(z). 



According to the general theory, the coefficients K t U) (z) are essentially deter- 

 mined except for certain arbitrary constants. Furthermore, the initial approximation 

 [K^iz), X 2 <°)(z), K 3 ^(z), J£ 4 (0) (z)] is uniquely determined up to a common 

 numerical factor. The actual calculation of the functions K^ (z) is, however, to be 

 done byi. solving certain differential equations and is in general very complicated. 



A practical method of obtaining the functions -K"i 0) can be devised by merely 

 using the structure of the solution (13). If we calculate the formal asymptotic expan- 

 sions of <&, for finite values of y — y c , by using the known expansions for u, we must 

 obtain asymptotic solutions expressible in terms of the solutions (8) and (9). Since 

 this can be done for four linearly independent solutions u, we have essentially four sets 

 of relations (for each order in A -2 ) for the determinations of K^ In the above argu- 

 ment, the constants of linear combinations are left undetermined. However, the regu- 

 larity condition for the coefficients K^^iz) is found sufficient for the determination 

 of all the constants to the extent specified in the general theory. Further details of the 



359 



