65 



number. The Squire transformation is most useful 

 for a two-dimensional boundary layer because the 

 velocity profile is unchanged. Thus all eigen- 

 values of three-dimensional waves can be obtained 

 from known eigenvalues of two-dimensional waves 

 with no additional calculations. In a three- 

 dimensional boundary layer, the velocity profile 

 must change and the Stuart transformation is pre- 

 ferred because the frequency can remain fixed at 

 a given Reynolds number as the phase orientation 

 angle iJj is varied. 



Spatial Stability Theory 



Statement of the Problem 



In the spatial stability theory, a and 3 are com- 

 plex and 0) is real. Neither transformation is of 

 much utility except when 



ct./E 



"r/Pr- 



(6) 



When (6) is not satisfied, a is complex, and in the 

 Squire transformation both R and o) are also com- 

 plex as well as U for a three-dimensional boundary 

 layer. In the Stuart transformation, U is complex 

 for all boundary layers. With complex quantities, 

 we might as well deal directly with (2) , as these 

 equations have already been reduced to fourth order 

 and nothing is to be gained from an additional trans- 

 formation. There only remains the question, to be 

 answered later in this Section, of whether any use 

 can be made of the simplification offered by (5) . 



It is convenient to define a real wavenumber 

 vector 



measure of the relative instability of different 

 velocity profiles, and its amplitude can be applied 

 to the transition problem. 



Introduction of an Amplitude Equation 



In order to describe wave propagation in the non- 

 uniform medium of the boundary layer, equations are 

 needed for the wave amplitude and the change in the 

 wavenumber vector in addition to the dispersion 

 relation. Even though no amplitude calculations 

 are included in this paper, a consideration of the 

 amplitude equation will help us select ip. 



In a nonuniform medium the elementary modes (1) 

 are not general enough and must be replaced by 



u(x,y,z,t) = A(x,z,t) exp[a (i())x]f (y)exp[i (a x 



+ B z - tot) 

 r 



(7) 



In this, the exponential amplitude factor has been 

 written separately in terms of the spatial amplifi- 

 cation rate a(iji). This amplification rate is the 

 magnitude of a (k,ii),u,x,z) considered as a function 

 of k,i(),a),x,z with a fixed value of ijj. Each if de- 

 fines a coordinate 



X = cosip X + sinii z 



along which the wave growth is directed. 



Nayfeh et al . (1978) have derived an equation 

 for the amplitude factor A(x,z,t) on the basis of 

 the multiple scales technique, with A considered 

 to be a slowly varying function of x,z,t, as are 

 a, 6,0) and f(y). In a uniform medium, and with A 

 independent of time, their equation reduces to 



3 A 3 A „ 

 C ^— + C ^— = 0, 

 X d X Z d Z 



(8) 



k = (aj.,Bj.) , 

 and a real spatial amplification rate vector 



where C = (Cx,C2) 

 We may note that 



is the (complex) group velocity. 

 (8) is also obtained from 



a = (-aj_, -e>^) , 



in place of the complex vector k - lo. The magni- 

 tudes of the vectors are k and o, and their di- 

 rections are given by the two angles 



i> 



tan ^ ( 



!/a^) , 



ip = tan ( B . /ct . ) 



Equation (6) is now seen to be a statement that 

 k and o are parallel (ip = if)) . Plane waves with 

 if 7^)J) have been termed inhomogeneous by Landau and 

 Lifshitz (1960). 



The solution of the eigenvalue problem set up 

 by (2) and (3) gives the complex dispersion relation 



3 t 



+ (V.C)A 



0, 



(9) 



which is the energy conservation equation of Whit- 

 ham's theory (1974). Davey (1972) has applied (9) 

 to non-conservative wave motion in a two-dimensional 

 mean flow, and refers to the amplitude function A 

 as a pseudo amplitude, or the 'dispersive part' of 

 the amplitude. 



Spatial Mode - Real Group Velocity 



We restrict ourselves first to the case of C real 

 and define the orthogonal coordinates 



to = fl (k,a,x,z) . 



Even with u, x and z fixed, there remain four real 

 wave parameters: k, if, a and if. Only two of 

 these can be determined in a single eigenvalue 

 calculation, e.g., k and a with if and if specified. 

 The angle if can be considered an independent 

 variable on the same basis as the frequency. The 

 problem is to choose if. What we are looking for 

 is a single spatial mode which serves the same 

 purpose as a two-dimensional spatial mode in a 

 two-dimensional boundary layer, where it represents 

 the wave produced by a stationary harmonic source . 

 " The amplification rate of this mode is used as a 



X = cosd) X + sinil) z, 

 gr ^gr *^gr 



z = -sinil) X + cosil) z, 

 gr gr gr 



(10a) 

 (10b) 



where 



ijj = tan ^ (C /C„) . 

 ^gr z X 



The angle if„j_ defines the direction of the charac- 

 teristic coordinate x^^^,, which is identical to a 

 group velocity trajectory, and A is constant along 

 each characteristic according to (8) . 



