A thermoacoustic approach for active noise control of a vibrating surface

We study the effect of surface heating on sound radiation by a vibrating boundary. Focusing on a setup of an infinite planar wall interacting with a semi-infinite expanse of a gas, the system response to arbitrary (small-amplitude) vibro-thermal excitation is investigated. Starting with the case of sinusoidal actuations, the superposed effect of boundary heat-flux excitation at a common frequency ω is examined. The entire range of frequencies is considered, where, depending on the ratio between ω and gas kinetic collision frequency ωcoll, fundamentally different flow regimes follow. The two limit cases of ω ≤ ω_coll (continuum-flow conditions) and ω ≥ ω_coll (ballistic flow regime) are investigated analytically, based on continuum equations and collisionless Boltzmann equation, respectively. In between, an intermediate interval of frequencies ω ∼ ω_coll is analyzed numerically, based on the direct simulation Monte Carlo method. In search for optimal conditions for boundary sound reduction, it is found that effective attenuation is obtained when boundary heat flux is applied at opposite phase to surface actuation. Amplitude-wise, conditions for minimization of the acoustic field vary between the limits: at low-frequency conditions, wave radiation extends over large distances from the wall, and optimal sound reduction is achieved when the ratio between wall-inserted thermal and kinetic energies \E_q/E_k\_opt = γ/γ - 1) with? denoting the ratio between gas specific heat capacities). At high-frequency conditions, wall signal affects only a thin gas layer of the order of the mean free path) in the vicinity of the boundary, and optimal attenuation is achieved when \E_q/E_k\_opt = 1. The analysis is extended to consider the system response to non-periodic excitations, for the specific case of a delta-function input. Making use of the above collisionless- and continuum-limit analyses, early- and late-time system responses are computed and rationalized. While conditions for optimal sound reduction at late times coincide with the low-frequency predictions in the periodic case, no counterpart agreement is found between early-time analysis and high-frequency periodic behavior.