66 



The amplitude portion of (7) is now 

 a(x,z) = A(z )exp[a(ii))x] , 



(11) 



and (7) can be interpreted as a certain type of 

 solution for a uniform medium when A is variable, 

 provided only that A is constant along a character- 

 istic. A knowledge of A along some initial curve 

 completely specifies a along the characteristics 

 of A, and the characteristics of A are also the 

 characteristics of a. Therefore we can write (11) 

 as _ _ 



a(Xgj.) = aQ(Zg^)exp{0gj.Xgj.), (12) 



where (10) has been used to eliminate x and 



Ogj- = a(iii)cos (ijj-iiigj.) . (13) 



Consequently, (7) becomes 



u(x,y,z,t) = a^Cz )exp(agj,Xgj,)f(y)exp 



wt)]. (14) 



normal to Xgj-, the growth in different directions 

 follows the usual vector law with the amplification 

 rate in direction iii given by 



a (ill, ) = a cos (i|), - ijj ) . 

 i gr '■ gr 



We can use (13) to (a) determine 



Ogj, from a(i|i) 



provided ij) is known; (b) determine $ if two 

 neighboring values of a(ii)) are known; and (c) answer 

 the question left open previously of whether we can 

 make use of the simplification in the spatial theory 

 afforded by (6). The latter is easily done. With 

 (6) , the transformation (5) applies to spatial waves 

 and gives 



-a. = a (ii) cosii/. 

 With i) = i), (13) relates a (^) to -a. by 



a cosil) 

 gr gr 



(15a) 



2- 



[a (iJ;')cosi|j] (1 + tanil) tanij) ) cos ip . (15b) 



If a,-, is a constant everywhere, the spatial mode 

 (14) represents a physical wave in the entire x,z 

 plane that could be produced by a particular 

 stationary harmonic line source in a uniform medium. 

 If a^ is constant only along a characteristic, we 

 have a form of ray theory, and (14) in turn applies 

 only along a characteristic (ray) . In other words, 

 X and z are constrained to follow the characteristic. 

 The latter viewpoint is more useful for a general, 

 nonuniform boundary layer, and also applies to a 

 stationary harmonic line source in a uniform bound- 

 ary layer when the locus and amplitude, distribution 

 of the source are arbitrary. 



Equation (13) was derived in M77 from a general- 

 ized Gaster relation between temporal and spatial 

 amplification rates. Its meaning can best be seen 

 from Figure 1, where the constant amplitude lines 

 for the two growth directions ijjgj; and if are shown. 

 These lines are normal to the direction of growth, 

 just as the constant phase lines are normal to the 

 direction of the wavenumber vector. A certain 

 growth along Xgj- in distance Ax requires the 

 amplification rate along x to be 1/cos (ijj-^gj-) 

 larger than the amplification rate along Xgj- to 

 yield the same growth along x in the shorter dis- 

 tance Ax = Axgj- cos (ip-ijjgj-) . It is this relationship 

 between a(!jj) and Ogj- that is expressed by (13). For 

 a fixed orientation of the constant amplitude lines 



It is evident from this expression that (6) is valid 



only for ijj =0 (or \p = ij^-j.) . However, a {^) can 



be used to calculate a , if ij;_„ is known , on the 



gr T 9^^ 

 same basis as any other o (v) ■ This procedure ^s 



obviously to be avoided when the direction of k is 



perpendicular to that of a . 



Spatial Mode - Complex Group Velocity 



With a complex dispersion relation, the group veloc- 

 ity, defined as 



C = 



3 ti) 

 da' 



3 (1) 



3 e 



(16) 



is also complex. For pure temporal or spatial modes, 

 C is real only at points of maximum amplification 

 rate. Consequently, it is important to know how 

 the complex C affects the preceding analysis. With 

 C and C complex, (8) is no longer hyperbolic, as 

 pointed out by Nayfeh et al . (1978). However, it 

 is still possible to proceed by defining a real 

 characteristic in the three-dimensional space (Xj- + 

 ixj^,z). Such a technique was used in a different 

 context by Garabedian and Lieberstein (1958) . 



The complex vector group velocity is conveniently 

 described in terms of a complex magnitude and a 

 complex angle by writing 



C = C cosip , C = C sinip , 

 X g z g 



where 



is the complex magnitude, and 



'\ = V * i*gi 



is the complex angle. 



The complex counterparts of (10) are 



X = cosil) X + sinii z, 



g g g 



(17a) 



(17b) 



(17c) 



FIGURE 1. Wave growth in direction JTg]. as described 

 by constant amplitude lines normal to Xgj- and to x. 



z = -sin'l' X + cosip z, 



g g g 



