Wave Propagation in Infinite Periodic Structures
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The wellknown BlochFloquet waves (BFW) are not the only manifestation of periodic structure waves (PSW) resulting from the scattering of structure waves (SW) in an infinite, uniform, one dimensional structure of equally spaced scatterers. For one structure wave type (SW1), forward transmission and backward reflection phase shifts are independent of wave propagation direction. For another structure wave type (SW2), the phase shifts have opposite signs for opposite directions of propagation. The differences in amplitudes and phases of the forward and backward SW within any “cell” between adjacent scatterers are equivalent to a continuous PSW amplitude ? convolved with a periodic structure function. Dispersion relations for PSW result from the solution of a quadratic equation for ? derived from imposing the same relative SW amplitudes and phases in all cells. However scatterer properties can be chosen that prevent PSW conformity with conservation of energy. This paper shows that in general two correlated PSW must coexist to achieve consistency with energy conservation. Dispersion relations for the two correlated PSW are derived for SW1 and SW2 wave types. BFW, where the SW1 backward reflection and forward transmission phase shifts differ by2/p±, are the unique case where conformity with energy conservation is achieved by a single PSW1. Other scattering phase shifts for PSW1, and all PSW2, require correlated pairs of PSW. Correlated pairs of PSW2 give the same results as a particular choice of phase shifts for correlated pairs of PSW1. An effect of correlated PSW pairs is that the SW reflectivity by the periodic structure is independent of the scatterer spacing similar to incoherent structure waves.
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