Primary resonance characteristics of a cylindrical bubble based on the multi-scale method

This paper describes a primary resonance theoretical model for a cylindrical bubble under acoustic excitation. Based on the multi-scale method, an analytical solution of the bubble–wall equation with second-order accuracy is obtained and numerically verified. The oscillation characteristics in the time domain and the frequency response characteristics of the oscillations under primary resonance are analyzed with different amplitudes and frequencies of acoustic excitation and the equilibrium radius of the bubble. This study yields the following primary findings: (1) For the cylindrical bubbles, the primary resonance of the bubble exists in unstable regions. Nonlinear behaviors such as jumps, hysteresis, and multivalued solutions may be widely present. (2) As the amplitude of the acoustic excitation and the bubble equilibrium radius increase, the backbone of the amplitude–frequency response curve bends to the left and the unstable region gradually expands. (3) When the dimensionless amplitude of the acoustic excitation is less than 0.005 and the bubble equilibrium radius is less than 1.0 × 10−5 m, the unstable region of resonance disappears.