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We develop a theory for acoustic-phonon transmission through a periodic superlattice with a finite number of periods. In particular, analytical expressions for the product of transfer matrices and phonon transmission rate are derived for the phonon propagation normal to the layer interfaces. The results are applied to the resonance phenomena of phonons in the triple-superlattice structure, i.e., the stack ABA of the periodic superlattices A and B. The phonon transmission rate and the resonance condition in this structure are also derived analytically. As a result, we show that the phonons in a frequency gap of the A superlattice can be transmitted significantly through the ABA superlattice structure when they satisfy the resonance condition. This happens even if the period of the A superlattice is infinity

Theory of acoustic-phonon transmission in finite-size superlattice systems