= = > >

Lemma 2 On the right side of the shock layer there is exactly one solution K. of (46} with a positive real part and two with a negative real part. Let r.p1 = ( 'Pl(l)> r.p1(2j)T be the vector corresponding to K.1 with Re(K.I) > 0. Then r.p1(2) ~ 0. Proof. Note that the corresponding constant coefficient half-plane problem,

has only one solution with Re(h:) ~ 0. Thus for Re(~) > 0, the solutions of (46) must also have this property. By Lemma 1, Re(K.;) ~ 0, and we have

( 49) The coefficient of K.3 in ( 46) is U ~ 0. Therefore the roots of ( 46) are continuous functions of i Since Re(K.;) = 0 is excluded by Lemma 1, (49) is valid for Re(~) = 0 also, and the first part of the lemma follows. Since r.p1 must satisfy

( ~ + K.1 U, K.1R)r.p1 = 0, we have r.p1(2) ~ 0, which completes the lemma.