67 



With X = X + ix. , and x required to be real, 

 r 1 g 



X = tanhili . (tanli) x - z) , 

 1 gi gr r 



and 



x„ = coshtp . I x 



g gi V gr 



tanilJ tanh^ip . z 

 gr gi gr 



With X real, the analysis for real C applies and 



g 



gives for the now complex amplitude along the real 

 characteristic , 



a(Xg) = a (z')exp[a{ii) cos (ij) - i)g)Xg]. (18) 



This expression differs from (12) in that_ipg is 

 complex, Xqj, has been replaced by z_ and z 

 (orthogonal to x'gj., see below). 

 We define 



gr 



X - tan<(' tanh^* , z 

 gr gr ^gi gr 



(19a) 



as the characteristic coordinate in the physical 

 plane to replace Xgj.. The angle between Xgj. and 

 Xgj. is given by 



tan(ii;^j. - ipgj-) = -tanifgj, tanh^ijlg^^. (19b) 



We can now write the complex amplitude (18) as 



a(Xg) = a^ (Zgj,)exp <; a(iiJ) 



cos (ijj - ijjgj.) COSh'^lJjgj^ 



+ i sindp - >()gj-) coshiJjgj_ sinh>Jjgj_ Xgj. 



ij Xgj 



(20) 



The real part of the exponential factor defines the 

 spatial amplification rate along x ' to be 



o{\i) ' cos(ip 



i) ) cosh ill . ■ 

 gr gi 



(21a) 



This expression differs from its real counterpart 

 (13), aside from the factor cosh^ijjgj^, in that here 

 ipgj. is the real part of the complex angle ijig and 

 not the angle formed by the real parts of Cx and 

 Cz- When if = ipg^' 



2 



and, unlike o 



=gr 



(21b) 

 a is not directly calculable as 



= cosh il) . 

 gr gi 



an eigenvalue; The imaginary part of the exponen- 

 tial factor of (20) gives the phase difference be- 

 tween the elementary mode growing along x and the 

 spatial mode (20) growing along x' . The phase 

 difference can be written as 



gr 



a' - a(il;)= a(ii) sin{ijj - i|) ) cosh>|) . sinhil; ., 

 gr gr gi gi ' 



(22) 



gr 



magnitude o given by (21b) . The eigenvalues are 

 preferably computed with ip = ^q^, but as ipgj- is 

 generally not known in advance, or for computation- 

 al convenience , they can be computed at a neighbor- 

 ing i> and a obtained from (21a) . If ip is sufficiently 

 close to <|;gr» the phase shift given by (22) is 

 negligible and the orientation angle if is unaffected 

 by the_transformation. 



If i>„ were independent of i|), both (14) and (23) 

 would also be expected to be independent of i> . How- 

 ever, as il departs from ijjqj- , o (ii) becomes large and 

 the evaluation of the complex derivatives in (16) 

 takes place in a region of the complex a and B 

 planes well removed from the points which give ii 

 The same difficulty exists in making comparisons 

 between temporal and spatial amplification rates . 

 Although the elementary modes with arbitrary i|/ are 

 available for the solution of an initial value 

 problem by superposition, we give physical signifi 

 cance here only to the special spatial mode with 

 iji = ijl . All of the other spatial modes , as well 

 as the combined temporal/spatial modes with a,6,(jJ 

 all complex, do not enter the present analysis 

 except for computational purposes. 



OBLIQUE WAVES IN A TWO-DIMENSIONAL BOUNDARY LAYER 



Numerical Example of Transformation Formulas 



We shall first discuss the transformations from 

 three- to two-dimensional form and then the trans- 

 formation between a spatial mode with arbitrary 

 growth direction and the mode with growth direction 

 ijjgj-. A single numerical example for the Blasius 

 boundary layer will suffice. We use the conventional 

 dimensionless frequency parameter F = (ij*v*/U*j, 

 and choose the length scale to be L* = (x*v*/U*)'^. 

 With this choice, the Reynolds number appearing in 

 (2) is R = (ut x*/v*)'2. The subscript 1 refers to 

 freestream conditions. 



For F = 0.2225 x lO""*, R = 1600 and ip = 50°, a 

 direct calculation of the eigenvalues with (6), i.e., 

 il = 50°, or the. completely equivalent two-dimensional 

 calculations with either the Squire or Stuart 

 transformations, gives 



k = 0.1671, oW = 4.119 X 10-2. 



Application of the wavenumber transformation rule 

 in (4) and (5) gives 



1 = 0.1074, -a. = 2.648 

 r 1 



10- 



for the complex wavenumber in the x direction^ 



If ijjqj- is computed in the neighborhood of i|i • 

 50° from (13) by means of the assumption that a 

 is independent of ii and with the frequency held 

 constant, we find 



gr 



where a is the wavenumber component in the Xg^ 

 direction. We can now write the complex C counter- 

 part to the pure spatial mode (14) as 



u(x,y,z,t) = a (z^j.)exp(ox' ) f(y) 



Fci' - a(ii;)'1x' - ut^ (23) 

 L gr J gr 



exp<i(a X + g z + 

 I r r 



With (23) we have arrived at the spatial mode 

 that will be used for the numerical calculations 

 to follow. The amplitude growth is along x'j. with 



il = 9.39° 



gr 



to be an approximate value for the real part of the 

 complex angle of the group velocity vector. (If 

 the wavenumber is held constant, ijjgr = 8.80°; a 

 value closer to the angle formed by the real parts 

 of Cx and C^-) The eigenvalues of the ij) - 50° wave 

 with il = 9.39° are 



k = 0.1669, o = 3.127 



gr 



10' 



•3 



