Nonlinear model of acoustical attenuation and speed of sound in a bubbly medium

The presence of microbubbles (MBs) in a medium changes the medium's acoustic properties and increases the attenuation of the bubbly medium. Current models of ultrasound attenuation in a bubbly medium are based on linear approximations; that is MB undergoes very small amplitude oscillations. Thus linear models of attenuation are not valid in many regimes used in diagnostic and therapeutic ultrasound applications. In this study, a model is developed that incorporates the nonlinear attenuation and sound speed by deriving the complex wave number from the Calfish model for the propagation of acoustic waves in a bubbly medium. Using the methods of nonlinear dynamics, we have classified the behavior of MBs for a wide range of frequencies and applied pressures. The results of the bubble oscillations are visualized using the bifurcation diagrams of the radial oscillations of the MBs as a function of the incident pressure. It is shown that depending on the frequency of the ultrasound wave, the nonlinear oscillations of the MBs can be classified into 5 main categories in which the MBs oscillations exhibit: 1. Linear resonance (fr), 2. Pressure-dependent resonance (fs), 3. Sub Harmonic (SH) resonance (fSH), 4. Pressure-dependent SH resonance (fpSH) and 5. Higher order SH resonance oscillations (fn). Results show that when MBs are sonicated by their fr, the effective attenuation of the medium can potentially decrease as the pressure increases, which is in good agreement with experimental observations. When sonicated with their fs, the effective attenuation of the medium is smaller than in the case of fr. This happens only below a pressure threshold that corresponds to the saddle node bifurcation in the corresponding bifurcation diagram. Above this pressure, the effective attenuation and sound speed increase abruptly by ~5 and ~2 folds, respectively. In the other classified sonication regimes (fSH, fsSH and fn) (3-5), the attenuation and sound speed changes are negligible below the pressure threshold corresponding to the SH oscillations. As soon as the pressure increases above the threshold for SH oscillations (e.g. period doubling in the bifurcation diagram), the effective attenuation increases abruptly (~ up to 3 fold), however the maximum exhibited attenuation is ~10 to 50 folds smaller than the maximum attenuation in case of sonication with fr and fs.