On matched asymptotic expansion solutions of a bucklewave propagation problem

In the present paper, the method of matched asymptotic expansions is used to derive approximate analytical solutions to a model problem related to bucklewave propagation in undersea pipelines. The model problem, introduced by Chater, Hutchinson and Neale (in: Collapse – The Buckling of Structures in Theory and Practice (Thompson, J.M.T., Hunt, G.W., eds.) Cambridge Univ. Press, Cambridge, 1983), consists of a linear elastic beam resting on a nonlinear elastic foundation, with inclusion of axial tension and inertia, related to a propagating buckle. When tension dominates over inertia, it is found that the asymptotic solution describing a propagating buckle has the character of nonoscillatory motion. When tension and inertia forces are of equal magnitude, or when inertia forces dominate over tension, the solutions are found to be oscillatory, but still localized around the bucklewave front. Finally, it is shown that when the propagation speed of the buckle exceeds a critical value, a spatial instability sets in, with exponentially growing amplitude of the beam deflection.