WKB Method with Repeated Roots and Its Application to the Buckling Analysis of an Everted Cylindrical Tube

For a linear ordinary differential equation of variable coefficients in which the highest order derivative $d^ny/dx^n$ is multiplied by a small parameter $\epsilon^n$ say, a WKB solution of the form $y(x)=f(x) \; {\rm exp}\; (\frac{1}{\epsilon}\int^{x} s(x) dx)$ can be sought. To leading order, s(x) satisfies an nth order algebraic equation. It seems that all the existing books on singular perturbation methods have discussed only the case when the roots of this algebraic equation are distinct except at possibly a finite number of points. In this case n independent solutions can readily be obtained and the general solution is a linear combination of these n solutions. When the algebraic equation has repeated roots, it is not immediately clear how to obtain n independent solutions. In this paper we first show, through a simple model problem, how the WKB method should be applied when double roots arise. We then apply the ideas to the WKB analysis of the buckling of an everted circular cylindrical tube. A sim...