Acoustic wave behavior on a finite 1 dimensional periodic lattice

Abstract

There are frequency regions which are called band gaps in periodic structures. If the frequency of the wave is in a band gap, then the wave cannot propagate through the structure. But it is valid only if the period of the structure is infinity. So it is important to understand the behavior of waves in finite periodic lattice for practical applications. In this research, the relations between the transmission coefficient and the periods of the periodic structure in band gap frequency regions are studied. The transmission coefficient of a finite periodic lattice exponentially decreases with respect to the periods of the structure and the base is obtained from the Floquet’s theorem. If there is a defect in the periodic lattice, the transmission coefficient shows a peak at the defect mode frequency. The peak value is determined by the difference between the periods of the lattices on the left and the right of the defect. The half power band width of the peak exponentially decreases with respect to the periods of the lattice. Also the effect of the lattice distortion is studied. If the length of the unit lattice has a random error, the transmission coefficient of the finite periodic lattice increase compared to the perfectly finite periodic lattice in the band gap frequency regions. And the 1-dimensional model is extended to the 2-dimensional models which are a multilayer system and a rectangular lattice. The transmission coefficient of the 2-dimensional finite periodic lattice also decreases exponentially with respect to the periods. The base is the function of not only frequency but also the incident angle. From the research the transmission coefficients of periodic structures is approximately obtained with respect to the spatial periods. Though the approximation formulas are derived from simple models, they can be applied to general structures because of the Bloch theorem.