<p>We consider a circular cylindrical elastic membrane tube which is initially at rest and subjected to a given constant axial extension. We then consider the acceleration waves arising from the boundary condition defined for the axial velocity at one end of the tube. Using the singular surface theory, we study the propagation of acceleration waves in the context of nonlinear elasticity. The temporal evolution and propagation speeds are determined for a general incompressible elastic material. We deduce the conditions which determine whether the amplitude of the longitudinal acceleration wave blows up or not. Considering some well-known examples of strain-energy function, our study demonstrates that different constitutive models behave very differently and an amplitude blow-up may occur depending on the magnitude of the initial stretch for some elastic materials. On the other hand, we show that when the state of the medium ahead of the longitudinal acceleration wave is its natural state, an amplitude blow-up does not occur.</p>