Nonreciprocal and even Willis couplings in periodic thermoacoustic amplifiers

Thermoacoustic amplifiers are analyzed in the framework of nonreciprocal Willis coupling. The closed form expressions of the effective properties are derived, showing that an applied temperature gradient causes the appearance of a nonreciprocal Willis coupling. Even and nonreciprocal Willis couplings are exhibited already in the first-order Taylor expansion of the solution and are of equal modulus but opposite sign, thus suggesting that the even Willis coupling is a reaction to the nonreciprocity introduced by the temperature gradients. These Willis couplings cause a coalescence point in the k space, which deviates from Re(k)=0 (with k the wave number) and is thus a zero-group-velocity point, as well as the opening of an amplification gap at low frequency. Effective parameters and scattering properties are found in excellent agreement with experimental results. This article paves the way to further control the acoustic waves at very low frequencies with nonreciprocal systems.