Homogenization of one-dimensional oscillatory systems with spatially modulated parameters

Oscillations of one-dimensional systems with spatially modulated parameters, in particular, strings with a varying cross section, are investigated. The method of direct separation of the motions is used, adapted to investigate systems in which separation of the variables is carried out in the spatial coordinate, rather than in time. It is established that modulation of the cross section of the string leads to the occurrence of a spectrum of additional high eigenfrequencies, which correspond to small wave numbers. A simple physical explanation of this effect is proposed. Analytic expressions are obtained for the eigenforms and eigenfrequencies of the oscillations of the periodic structure considered. It is shown that the modulation of the cross section of the string can be used to monitor the nature of its oscillations, in particular, suppress high frequencies.