This theoretical study of the effect of viscoelastic material properties of a pipe conveying one-dimensional, non-stationary flow shows that, in addition to exponential attenuation of wave fronts, the disturbance is transformed into a diffusion front whose thickness increases as √t, and which propagates at a velocity corresponding to the retarded elasticity of the pipe material [E(∞)]. The behaviour of pressure disturbances in the special case of water-hammer resulting from sudden closure of a valve is illustrated by a numerical application based on the characteristics method.