High order harmonic balance formulation of free and encapsulated microbubbles

The radial responses of free and encapsulated microbubbles excited by an ultrasonic plane wave with a large wavelength in comparison with the bubble size are governed by NonLinear Ordinary Differential Equations (NL-ODEs). The nonlinear frequency response gives the harmonic content of the time response and constitutes the expected outcome of a high order harmonic analysis. In this paper, high order harmonic balance analysis of modified “RPNNP” (bubble), Hoff and Marmottant (contrast agents) models is performed with an open-source software program. For this purpose, the original NL-ODEs are recast into nonlinear systems in which the nonlinearities are at most quadratic. In the spectral domain, this recast provides close form and aliasing-free solutions of arbitrarily large numbers of harmonics. Relevant quantities such as primary and secondary resonances and the nonlinear amplitude threshold of the excitation wave are evaluated. The frequency curves drawn up characterize the bending and quantify the jump frequencies and amplitudes of each harmonic component. The results obtained with this predictive method confirm that it should provide a useful tool for nonlinear bubble detection and sizing and for contrast agent designing.