In the periodic motion of the blood flow in the circulatory system, the generated pulse wave at arterial walls plays a vital role to accelerate or decelerate the motion. In this article, the pulse wave propagation of arterial blood in the viscoelastic arteries is studied. The constitutive fluidwave hyperbolic system of equations is derived for the incompressible laminar blood flow in a cylindrical shaped tube. The blood is assumed to follow Newtonian characteristics while the arterial wall obeys viscoelastic characteristics which is characterized by modified Zener model. The derived equations are numerically solved by Dumbser-Osher-Toro (DOT) based Riemann solver with weighted essentially nonoscillatory (WENO) discretization scheme. The results are found to be in line with the earlier simulated clinical trials. It is observed that during blood flow, the wall pulse wave gets damped faster in the viscoelastic wall as compared to elastic wall and predicts the same nature as observed clinically for the physiologic blood-wall interaction to measure the proper functioning of the cardiovascular system.