Wave propagation in infinite periodic structures taking into account energy absorption

Abstract

This paper explores the possibility of generalised periodic structure waves (PSW) that include the well-known Bloch-Floquet (BF) waves as a special case. We consider two types of structure waves (SW) in an infinite, uniform, one dimensional structure of equally spaced scatterers that also absorb energy. For the first structure wave type (SW1), forward transmission and backward reflection phase shifts are independent of wave propagation direction. For a second structure wave type (SW2), the phase shifts have opposite signs for opposite directions of propagation. Examples of SW1 are bending waves, such as flexural waves of a plate, and for SW2 longitudinal waves, such as acoustic waves in a fluid. The differences in amplitudes and phases of the forward and backward SW within any "cell" between adjacent scatterers are found to be equivalent to continuous PSW convolved with a periodic structure function. Finding the PSW dispersion relations requires a function that is the solution of a quadratic equation derived from imposing the same relative SW amplitudes and phases in all cells. Conservation of energy identifies physically acceptable PSW. For no energy absorption and backward and forward scatter phase shifts differing by±p/2, PSW of the first type (PSW1) are BF waves that propagate unattenuated in passing bands and are evanescent in stopping bands. Including energy absorption for the same phase shifts, PSW1 propagation occurs at all wavenumbers but is attenuated. This extends the BF dispersion relations to include energy absorption which blurs the distinction between passing and stopping bands. For other scatterer phase shifts, PSW1 may still be possible but only at discrete wavenumbers. In contrast PSW of the second type (PSW2) are only consistent with conservation of energy at discrete stopping wavenumbers that are the Bragg reflection condition. PSW1 also exhibit Bragg reflection, but as a narrow stopping band for small scatterer reflectivity and energy absorption. A theory for incoherent wave energy scattering in an infinite periodic structure is also developed, and its results for energy reflection, transmission and absorption are similar to those of PSW1 except for coherence effects.