A. S. Iberall 



We lose very little — for the present purposes — to neglect the difference 

 between [d^ - \ - joj] and [d^ - x - jm^] and [d^ - \ - jm^ + jSR„„(p]. Thus, we 

 will assume (a^ = - jw - \ ). (The difference is first order in (low) Mach number.) 



[D2 + a2]U + qD [DU+ j Va V+ jSX] - DP = . 

 [D^ + a2] V + jaq [DU+ iK/a V+jSX] - jaP = , 

 [D2 + a2]X ^ -R„„q)'U , 



[DU+ jVa V+ jSX] - 7/32 gX _ yB/3^g/a V + 7/32 j^P - jB^ jcoT = , 



1/a [D2+ a2 + j (i-a) w] T - 2ejSR^„q)'U - [2eR„^cp 'D + (7 - 1) g] X 



-S/a [2eR„^(p'D+(7-l) g] V + (7-I) j«P - , 



i.e., we will consider that the only salient perturbation comes from qp'. 

 Examine this. 



In the Core (cp' % 0) 



[D2+ a2]U + qD [DU + j X/a V+ jSX] - DP = , 



[D2 + a2] V + jaq [DU+ jA./a V+ jSX] - j aP = , 



[D2 + a2] X = , 



[DU+ jVa V+ j8X] - 7/32 gX - 7S/32g/a V + y/3^ jcoP - ft^jcvT = , 



1/a [D2 + a2+ j (1-a) w] T - (7- 1) gX - S/a (7- 1) g V + (7- 1) jojP = 0. 



Except for the indifferent replacement of M^, by co, this is the same as Eqs. (8). 



Thus (to first order — anticipating the final results) two independent sym- 

 metry solutions emerge: 



^10 ^ J "'^^o ^^" ax + /V(1-ct) wBq sin ax - /3^d C sin dx 



+ jbD sinh bx , 

 Vjjj = ""^^^0 "^^^ ax + j ( 1 - cr) aacoB^ cos ax + ja/32C cos dx 



-aDg cosh bx , 

 XjQ = SaAg cos ax , 

 TjQ - cr(y- 1) SagBg cos ax + ctCq cos dx 



+ (7- 1) coD^ cosh bx 

 PjQ = jq (7- cr) Sag/32a)Bp cos ax - j (1 - a- aq) /3^<^C^ cos dx 



+ wD cosh bx 



742 



