The method of reverberation-ray matrix (MRRM) combined with the Floquet–Bloch theorem, which serves as an alternative method, is presented for accurately analyzing longitudinal waves in general periodic multiphase rods. Closed-form dispersion relation of periodic quaternary rods is derived. Based on this relation, the functions of constituent-rod parameters in the formation of longitudinal-wave band structures are analytically revealed. Numerical examples validate the proposed method and indicate the characteristics/applications of all kinds of dispersion curves that include the frequency-wave number spectra, the frequency-wavelength spectra, the frequency-phase velocity spectra, the wave number-phase velocity spectra and the wavelength-phase velocity spectra. The effect of unit-cell layout on the frequency band properties and the functions of constituent-rod parameters in the band structure formation are also illustrated numerically. The analysis and interpretation of longitudinal waves in periodic multiphase rods given in this paper will push forward the design of periodic structures for longitudinal wave filtering/guiding and vibration isolation/control applications.

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