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.

We present a novel formulation of the Rayleigh–Plesset equation to describe stable gas bubble dynamics in viscoelastic media, using bubble volume variation, rather than radius, as the primary variable of the resulting nonlinear ordinary differential equation. This formulation incorporates the linear Kelvin–Voigt model as the constitutive relation for the surrounding fluid, capturing both viscous and elastic contributions, to track the oscillations of a gas bubble subjected to an ultrasonic field over time. The proposed model is solved numerically, subjected to a convergence analysis, and validated by comparisons with theoretical and experimental results from the literature. We systematically investigate the nonlinear oscillations of a single spherical gas bubble in various viscoelastic environments, each modeled with varying levels of rheological complexity. The influence of medium properties, specifically shear elasticity and viscosity, is examined in detail across both linear and nonlinear regimes. This work improves our understanding of stable cavitation dynamics by emphasizing key differences from Newtonian fluid behavior, resonance frequency, phase shifts, and oscillation damping. Elasticity has a pronounced effect in low-viscosity media, whereas viscosity emerges as the dominant factor modulating the amplitude of oscillations in both the linear and nonlinear regimes. The model equation developed here provides a robust tool for analyzing how viscoelastic properties affect bubble dynamics, contributing to improved the prediction and control of stable cavitation phenomena in complex media.

We present a novel formulation of the Rayleigh–Plesset equation to describe stable gas bubble dynamics in viscoelastic media, using bubble volume variation, rather than radius, as the primary variable of the resulting nonlinear ordinary differential equation. This formulation incorporates the linear Kelvin–Voigt model as the constitutive relation for the surrounding fluid, capturing both viscous and elastic contributions, to track the oscillations of a gas bubble subjected to an ultrasonic field over time. The proposed model is solved numerically and validated by comparison with theoretical results from the literature. We systematically investigate the nonlinear oscillations of a single spherical gas bubble in various viscoelastic environments, each modeled with varying levels of rheological complexity. The influence of medium properties, specifically shear elasticity and viscosity, is examined in detail across both linear and nonlinear regimes. This work improves our understanding of stable cavitation dynamics by emphasizing key differences from Newtonian fluid behavior, resonance frequency, phase-shifts and oscillation damping. Elasticity has a pronounced effect in low-viscosity media, whereas viscosity emerges as the dominant factor modulating the amplitude of oscillations in both the linear and nonlinear regimes. The model equation developed here provides a robust tool for analyzing how viscoelastic properties affect bubble dynamics, contributing to improved prediction and control of stable cavitation phenomena in complex media.

Understanding the dynamics of gas bubbles in viscoelastic media is crucial for applications involving stable cavitation under ultrasound, such as drug delivery, materials processing, and biomedical imaging. The Rayleigh-Plesset equation formulated in terms of bubble volume variation, incorporating viscoelastic effects via the linear Kelvin–Voigt model, is extended here to a fourth-order approximation. This formulation allows a more accurate description of nonlinear bubble dynamics at finite acoustic amplitudes. The resulting equation is solved numerically under various acoustic conditions, with particular emphasis on driving frequencies near the bubble’s resonance and differences between Newtonian and viscoelastic media. To identify the physical conditions under which higher-order nonlinearities become necessary, a decision-tree classification analysis is performed. The results show that the proximity to resonance and the excitation amplitude are the primary determinants of higher-order nonlinear effects, while rheological properties act as modulators, with viscosity exerting a stronger influence than elasticity within the explored ranges. This work provides a physically interpretable criterion for selecting the appropriate model order, improving the prediction and control of nonlinear bubble oscillations under ultrasound excitation in viscoelastic media.