We obtained the time dependence of the transverse profile, veloeity and energy of longitudinal solitary waves in an elastic rod , in the case for which the viscoelastical damping is small but cannot be neglected ,by using the method of perturbation. The results showed that owing to the presence of viscoelastical damping, the height and velocity of solitary wave diminish, while the width increases with time and the energy dissipates gradually until it disappears.