<p>Asymptotic approximations and stability of shock-like propagating bucklewave solutions are investigated for a model problem related to bucklewave propagation in undersea pipelines. The model problem, introduced by Chater, Hutchinson, and Neale (in: Thompson, Hunt, eds) Collapse - the buckling of structures in theory and practice, Cambridge University 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. The beam is subjected to a constant lateral load, representing a constant hydrostatic pressure. It is shown that, at the so-called Maxwell load by which quasi-static bucklewave propagation is possible, the governing equation can be transformed into the stationary extended Fisher–Kolmogorov equation which, in turn, is equivalent to the canonical equation of Peletier and Troy (Spatial patterns: higher order models in physics and mechanics, Springer, New York, 2001). An asymptotic approximate solution is developed for the case where the shock-like bucklewave is oscillatory but in the form of an evanescent wave. The linear stability of this solution is verified.</p>