On the basis of the Kramers-Kronig relations and the dispersion equation the connection between the real and imaginary parts of the complex refractive indices for eigenwaves in weak spatial dispersion media is proposed. The theoretical description was applied to an experimental analysis of the amplitude-phase measurements of transmission spectra of semiconductor crystals near the exciton resonances. Two possibilities have been considered. The first is the detection of optical properties of normal waves diverted in space or in time. The second possibility refers to the analysis of the normal wave interference for plane-parallel crystals. It was shown that taking into account the interference of supplementary waves enables one to formulate the general concept and to derive the additional Kramers-Kronig dispersion relations and the dependence of the integral exciton absorption coefficient of the damping constant resulting from the influence of the spatial dispersion.