Nonlinear Ultrasonic Propagation in Bubbly Liquids: A Numerical Model

In this paper, we investigate the problem of ultrasonic propagation in liquids with bubbles. A new numerical algorithm is constructed to solve the acoustic field-bubbles vibration coupled system. For this purpose, a second-order equation written in a volume formulation is considered for bubbles vibration and coupled with the linear nondissipative wave equation, i.e., attenuation and nonlinear effects are supposed to occur exclusively because of the presence of bubbles. Nonlinear characteristics of the phenomenon are particularly analyzed and illustrated. Plane harmonic waves are first considered in a mixture of air bubbles in water, and conclusions about changes in the wave speed, attenuation, harmonic distortion, effective nonlinearity parameter and nonlinear effects with distance are given. In particular, a law relating the second-harmonic progression with the density of bubbles is found. The propagation of plane pulses is also analyzed to give results on nonlinear attenuation, changes of frequency, and self-demodulation. The influence of the resonance frequency of bubbles on the nonlinear field is then determined. Differences and similarities with nonlinear acoustics in homogeneous fluid are shown and commented. The possibilities and limits of an equivalent nonlinear fluid are then discussed. The propagation of a high-frequency pulsed signal in a bubbly liquid used in a biological application is also the subject of numerical experiments, for frequencies near and beyond the resonance frequency of the bubbles.