On the efficiency of a monopole source at noncontinuum conditions: Revisiting the pulsating sphere problem

The pressure field of a pulsating sphere is a canonical setup in acoustics, used for analyzing the acoustic efficiency of a monopole source at continuum flow conditions. We revisit this problem to characterize monopole radiation at non-continuum conditions, for which the Knudsen number (Kn) - marking the ratio between the molecular mean free path or time, and the characteristic length or time scales, respectively - is non-zero. The acoustic field generated by harmonic small-amplitude normal-to-wall displacements and heat-flux excitations is considered. The problem is analyzed in the entire range of gas rarefaction rates and excitation frequencies. Numerical calculations are carried out via the direct simulation Monte Carlo method, and are accompanied by analytical predictions in the ballistic- and continuum-limit regimes. Comparing with the reference inviscid continuum (Kn → 0) solution, the results characterize the impact of gas rarefaction on the acoustic source radiation. At near-continuum conditions, the pressure field combines three exponentially decaying components, reflecting thermoviscous and higher-order rarefaction effects, and yielding the familiar 1/r decay in the inviscid limit. Significantly stronger attenuation is observed in the ballistic limit, where boundary curvature results in “geometric reduction” of the molecular layer affected by the source, and the signal vanishes at a distance only few acoustic wavelengths away from the boundary.