RESUMEN

For a linear ordinary differential equation of variable coefficients in which the highest order derivative is multiplied by a small parameter epsilon 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 simple asymptotic expression for the critical ratio of the inner radius to the outer radius is obtained. Excellent agreement between the asymptotic and numerical results is found over almost the whole mode-number regime. We also deduce from the explicit expressions for the buckling modes that wrinkles are confined to a thin layer near the inner surface.