A Multiscale Analysis of Gas Bubble Oscillations: Transient and Steady-state Solutions

The nonlinear oscillations of a spherical bubble in an incompressible, viscous, liquid, subject to an acoustic pressure field, are investigated by means of a multiscale perturbation method. Approximate analytical solutions for the transient and the steady-state oscillations in the regions of the main resonance, first and second subharmonic and ultraharmonic are obtained at second order expansion. Through this method, simple analytic expressions for the frequency response curve are available. These allow a quantitative estimation of the maximum oscillation amplitude corresponding to a given excitation intensity. Furthermore, the thresholds for the excitation of the two subharmonic oscillations has also been obtained. The results exhibit a good agreement with numerical simulations.