In the article a methodology for researching wave processes in weakly nonlinear homogeneous one-dimensional systems - elements of machine and building structures is proposed. It is based on D'Alembert's method of constructing solutions of wave equations and the asymptotic method of nonlinear mechanics. Resonant and non-resonant cases are considered. In the non-resonant case, the influence of viscoelastic forces and small periodic disturbances on the oscillatory process is taken into account. In the resonant case, the influence of the disturbing force and the phase difference of natural oscillations is considered. Numerical methods for linear differential equations were used to analyze the obtained differential equations.