The paper is devoted to the analytical solution of the problem for 1D hemodynamical equations with periodic boundary conditions. The method of the solution is based on the asymptotic expansions on the small parameter and Fourier method. The attention is focused only on the first-order terms in the expansion. The solution, obtained for the particular case of initial conditions, is used for the comparison of rheological models of blood. It is demonstrated that the strongest damping takes place for the Power Law non-Newtonian model.