The flow of a Maxwell fluid in a pipe generated by a pulsating pressure gradient is considered. The problem is governed by two non-dimensional parameters, the Womersley number cc and a Deborah parameter gamma. We obtain a simple approximate solution in the limit of large a (pipe diameter much larger than the viscous penetration depth) by solving the resulting boundary-layer problem through a two scales perturbation technique, which also helps to understand qualitatively the structure of the flow as gamma increases. This limit is of interest, for instance, for the blood flow in arteries. The main qualitative differences between the Newtonian fluid flow (gamma = 0) and the highly elastic fluid with gamma = 0(1) or larger are easily discussed from the found solution in terms of just two algebraic functions of gamma. (C) 2015 Elsevier B.V. All rights reserved.