This paper studies the nonlinear resonance of a cavity filled with a nonlinear biphasic medium made of a liquid and gas bubbles at a frequency generated by nonlinear frequency mixing. The analysis is performed through numerical simulations by mixing two source signals of frequencies well below the bubble resonance. The finite‐volume and finite‐difference based model developed in the time domain simulates the nonlinear interaction of ultrasound and bubble dynamics via the resolution of a differential system formed by the wave and Rayleigh–Plesset equations. Some numerical results, consistent with the literature, validate our procedure. Other results reveal the existence of a frequency shift of the cavity resonance at the difference‐frequency component, which rises with pressure amplitude and evidences the global changes undergone by the bubbly medium under finite amplitudes. Finally, this work shows the enhancement of the amplitude of the difference‐frequency component generated by parametric excitation using the nonlinear resonance shift, which is more pronounced when the second primary frequency is constant, the first one is varied to match the nonlinear resonance, and both have the same amplitude.

A mathematical model that describes nonlinear ultrasonic wave propagation in bubbly viscoelastic media is developed by coupling the acoustic wave equation with a modified Rayleigh-Plesset equation, formulated in terms of bubble volume variation and incorporating the linear Kelvin-Voigt viscoelastic model. This formulation enables direct analysis of the influence of viscoelastic properties under soft tissue conditions. The differential system is numerically solved to investigate the transition from linear to nonlinear regimes in representative viscoelastic fluids and media that vary in shear elasticity. Laws of harmonic amplitudes vs. excitation at the source are defined from their polynomial fits. They confirm this nonlinear trend and reveal the influence of elasticity. To quantify these effects, the nonlinear parameter β is computed using a finite amplitude method. Our results show that increasing shear elasticity significantly attenuates nonlinear propagation and suppresses harmonic generation, with β decreasing as the shear modulus increases. In contrast, viscosity exhibits only a minor influence within the range studied in this work. These findings demonstrate that β is highly sensitive to the mechanical properties of the medium and can serve as an effective indicator to characterize the nonlinear acoustic response of bubbly viscoelastic media. The agreement with previous studies presents our new model as a valuable tool for the study of nonlinear ultrasound in bubbly soft tissues and materials.

Several numerical models have been developed in different configurations to simulate the behaviour of finite-amplitude ultrasound when interacting with tiny gas bubbles in a liquid. Since this interaction is highly nonlinear, specific models must be developed to understand the propagation of the waves in this kind of dispersive media for which their nonlinear and attenuation coefficients, as well as the sound speed, are extremely dependent on the ratio of the driven frequency to the bubble resonance. The bubble volume variation is mathematically modelled in the time domain through a Rayleigh-Plesset equation with terms up to the second order, whereas the time-dependent acoustic field relies on the wave equation in one or several dimensions. Both differential equations are coupled and auxiliary conditions are imposed. The differential systems are solved by the developed numerical models. In this paper we study in a three-dimensional resonator with axial symmetry how new harmonics obtained by nonlinear distortion can be enhanced by taking the nonlinear resonance effect into account, and we show that the generation of new frequency components by nonlinear frequency mixing exists. We also analyse the stable cavitation phenomenon in a three-dimensional focused field with axial symmetry by considering a nonlinear dependence of bubble generation in the liquid and the existence of primary Bjerknes forces.