Asymptotic analysis of a fourth-order turning-point problem in hydrodynamic stability

An asymptotic analysis is made of the so-called Pekeris modes of the Orr-So mmerfeld problem for plane Poiseuille flow. These are damped modes of the ' centre' or 'fast' type for which c(r) up arrow 1 and c(i) up arrow 0 as alp ha R --> infinity. The numerical results obtained by Orszag (J. Fluid Mech. 50 (1971) 689) for alpha = 1 and R = 10 000 showed that the eigenvalues fo r the even and odd modes of this type are very nearly equal, and one of the goals of this paper is to provide an analytical explanation for this rathe r striking result. Under certain simplifying assumptions we are led to a fo urth-order equation which can be viewed as a generalization of Weber's equa tion for the parabolic cylinder functions. The eigenvalue problem is then p osed on an infinite interval and we find that the eigenvalue relations for the even and odd modes are indeed the same even though the underlying analy sis is significantly different in the two cases. Explicit results are also given for both the even and odd eigenfunctions; The even eigenfunctions are similar to the Whittaker functions D-n(x) but the odd eigenfunctions invol ve Dawson's integral and certain polynomials.