Mathematical models for the acoustic response of a solids-loaded encapsulated bubble.

A method of active acoustic resonance interference spectroscopy is introduced for estimation of bubble properties. A modified form of Rayleigh-Plesset equation for forced oscillation of either a single free bubble or elastic shell encapsulated microbubble with attached solids loading is solved by the regular perturbation method for steady oscillatory solutions as a result of small amplitude acoustic excitation by a point sinusoidal oscillator. A model for the total pressure field at an acoustic receiver in an incompressible liquid is then solved by the regular perturbation method. Closed-form analytical solutions are found for pressure power at the acoustic receiver as a function of the excitation frequency and strength; the properties of the bubble, liquid, and encapsulating shell; and the geometry of the active monitoring system. The receiver pressure power exhibits a maximum due to bubble resonance and a minimum due to destructive interference between source and bubble response pressure fields at higher excitation frequencies. The inverse problem is solved to derive unique closed-form analytical estimators for bubble equilibrium size, attached solids mass loading, and encapsulating layer dilatational viscosity as a function of the frequencies of the fundamental resonance maximum, interference minimum, second harmonic maximum total average acoustic power, monitoring system, and phase properties.