The waveguide and resonance properties of inhomogeneous penetrable one–dimensional–periodical structures that consist of two different media are studied within the framework of a one–dimensional approximation. The pass and stop bands are determined. A dispersion relation for all the waveguide modes is obtained. Explicit expressions for low waveguide frequencies and corresponding phase velocities of waveguide modes for mono– and polydisperse media are found. The influence of the polydispersity of the sizes of heterogeneities on the low frequencies of a pass band is considered. A pass band in the range of low frequencies is detected. It is shown that the polydispersity does not affect the waveguide properties of a medium at low frequencies of the first pass band. The resonance phenomena in periodical media and structures are investigated. The resonance phenomena are shown to occur for an unlimited discrete set of frequencies if the group velocity of the waveguide mode for them is zero; in this case, the growth of the oscillation amplitude is localized in the neighborhood of a source (localization of the resonance). A synchrophasotron resonance at which the infinite chain of oscillation sources has the oscillations phase of a corresponding traveling wave from the pass band is detected.
