Abstract <p>This article is devoted to an in-depth study of the dynamic behavior of elastic rods with a constant cross-section, with an emphasis on the presence of elastic waves arising within these rods. Detailed mathematical models describing the motion of rod points over time are presented. These models take into account both conservative and non-conservative loads, providing a more complete understanding of the rod’s dynamics under various conditions. An important aspect of this work is adherence to fundamental conservation laws, such as the law of conservation of energy and the law of change of momentum. These fundamental principles of physics ensure the adequacy and realism of the obtained results, as well as their compliance with the fundamental laws of mechanics. The analysis revealed that the wave front arising in the rod moves along its length at a constant velocity. Moreover, the accelerations of all points on the rod during this motion remain equal to zero, indicating that the motion occurs at a constant, albeit time-varying, velocity. Furthermore, an interesting feature was discovered within the framework of transient dynamics: for the first natural frequency of oscillations, there exists an infinite number of linearly independent oscillation modes. This differs significantly from classical theory, in which each frequency typically corresponds to a single oscillation mode. This discovery opens new horizons for further research in the field of elastic beam dynamics.</p>