This paper deals with the study of the propagation of longitudinal waves in materials with point defects. The problem is described by the system of differential equations comprising the dynamic equation of the elasticity theory and kinetic equations of defect density allowing for the mutual influence of defects and a propagating wave, as well as the mutual recombination of defects. There are considered herein both the limiting cases presenting materials with one type of point defect (vacancies, interstitials) and the general case, if the material contains both types of point defects (vacancies and interstitials). A self-consistent mathematical model under conditions of the static deformation is reduced to a nonlinear evolution equation which combines the well-known wave dynamics equations of Korteweg-de Vries-Burgers and Klein-Gordon. Exact analytical solutions to the resulting evolution equation were found and analyzed. There was analyzed the influence of point defect parameters, which characterize the diffusion of defects, the rate of their recombination at sinks and the change in the volume of the material, if a single point defect is formed therein, on the harmonic wave amplitude and velocity. It was shown that in vacancy-containing media the low-frequency longitudinal waves have a greater amplitude and velocity than in interstitials-containing media. And in vacancy-containing media the low-frequency perturbation velocities reach larger values, but in interstitials-containing media they reach lower values as compared to high-frequency perturbations. A frequency range was identified wherein the dispersion of longitudinal waves was significant; in vacancy-containing media it was normal, and in interstitials-containing media, it was anomalous. An increase in the diffusion coefficient or a decrease in the dilatation parameter value contributed to the lower grade dispersion. For high-frequency waves, media with vacancies and interstitials were barely distinguishable; the availability of any point defects had almost no effect on the propagation velocity of high-frequency perturbations and their amplitude. It was noted that the defect diffusion coefficients did not affect the existence of an additional low-frequency wave in media containing two types of point defects. In the limiting case associated with the Korteweg-de Vries-Burgers equation it was shown that there were stationary waves with a decreasing or increasing profile for the function of the volume concentration of point defects. In the limiting case associated with the Klein-Gordon equation, with a negative value of the coefficient for the linear term with a zero derivative the propagation of stationary compression waves was possible with any sign of the coefficient for the nonlinear term; with a positive value of the coefficient for the linear term with a zero derivative, the propagation of a rarefaction wave was possible. In media with single type of point defect, the propagation of nonlinear stationary deformation waves is possible. Their velocity is fixed and its values differ for vacancy-containing media and interstitials-containing media, i.e. velocities pertain to different intervals. The type of material defects affects the type of the stationary wave, so in vacancy-containing media a compression wave is formed and in interstitials-containing media a rarefaction wave is formed. In vacancy-containing media, a nonlinear wave has the higher velocity but the smaller amplitude and width as compared to a wave propagating in interstitials-containing media.