Accompanying the frequency shift of the nonlinear resonance of a gas bubble using a dual-frequency excitation

Abstract

An asymptotic method is applied to analyze the nonlinear oscillations of a gas bubble driven by a dual-frequency excitation. More specifically, the latter is considered as a combination of two neighboring, incommensurate frequencies and is treated as a nonstationary excitation. This implies that both amplitude and phase of the bubble response are slowly oscillating at the time scale of the frequency difference, thus leading to a regime of aperiodic oscillations. The approximate solution is successfully compared with numerical simulations and reveals the possibility of achieving larger bubble response amplitude compared to the monofrequency excitation for sufficiently high driving pressure and specific values of the frequency difference. The asymptotic theory captures the generation of additional spectral components coming from the nonlinear mixing of both driving frequencies. This mechanism is responsible for a global enhancement of the dual-frequency bubble response as it enables an energy transfer towards more efficient components which are successively tuned with the nonlinearly shifted resonance frequency of the bubble, thus limiting the saturation due to softening.