Multiple Bragg scattering in arbitrary periodic acoustic waveguides

Periodic acoustic waveguides allow the propagation of only certain bands of frequencies, thus making them very effective acoustic filters. The particular distribution of these bands depends critically on the waveguide geometry. If the variation of the waveguide walls happens to be sinusoidal, then the waveguide preferentially reflects only the frequency which matches the Bragg condition for complete backscatter. The reflection of a single frequency in the sinusoidal waveguide seems to contrast the case of a more general periodic structure, where numerous bands of frequencies are disallowed. In this work, the relationship between the sinusoidal waveguide and an arbitrary periodic waveguide is studied using a transmission line model. Inspection of the input reflection coefficient as a function of frequency yields striking similarities to the Fourier transform of the wall variation, implying that a waveguide whose walls are defined by an arbitrary periodic function independently Bragg reflects the individual Fourier components of the wall variation function. The specific example of a waveguide with a square wave variation (a waveguide with periodic expansion chambers) will be discussed and compared to the literature, as well as the possibilities of recovering the waveguide geometry from the input reflection coefficient.