Wave propagation and attenuation in an infinite periodic structure of asymmetric scatterers

Periodic structures exhibit interesting periodic structure wave (PSW) phenomena although the literature is mostly devoted to symmetric rather than asymmetric systems. Waves in asymmetric structures is a topic of interest for applications where the energy flow needs to be reduced in particular directions. Relatively little has been investigated about wave modes in asymmetric periodic structures that exhibit nonreciprocal wave propagation and attenuation. This paper reports on such an investigation but restricted to an infinite one-dimensional structure of equally spaced nonreciprocal scatterers of structure waves (SW). An infinite structure without boundary effects has wave characteristics the same between every pair (i.e. "cell") of adjacent scatterers, which simplifies the theory to considering only two adjacent cells. It is found that only one type of wave mode, an incoherent energy wave (IEW), can exist in an infinite nonreciprocal periodic structure. In the case of elastic scattering the IEW is a "passing" band in one direction and a "stopping" band in the opposite direction. The IEW is also an allowed mode for symmetric scatterers. However for symmetric scatterers three other modes are also possible for both directions. One is the well-known Bloch-Floquet wave (BFW), which for elastic scattering alternates between passing and stopping bands as a function of wavenumber. The other two "non-BFW" modes result from symmetric moduli of the reflection and transmission scattering coefficients, but asymmetric scattering phase shifts. In the case of elastic scattering one non-BFW is a passing band and the other is a stopping band. In contrast to BFW resulting from multiple reflections and transmissions of a single SW, the IEW and two non-BFW wave modes require two different but correlated SW coupled by scattering