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The small-amplitude radial oscillations of a gas microbubble encapsulated by a viscoelastic solid shell and surrounded by a slightly compressible viscoelastic liquid are studied theoretically. The Kelvin–Voigt and 4-constant Oldroyd models are used to describe the viscoelastic properties of the shell and liquid, respectively. The equation for radial oscillation is derived using the method of matched asymptotic expansions. Based on this equation, we present the expressions for damping coefficients and scattering cross sections at the fundamental frequency and at twice that frequency. The numerical maximization of the amplitude-frequency response function shows that the resonance frequency for the encapsulated microbubble highly depends on viscous damping, and therefore, significantly differs from the undamped natural frequency. The effects of the shell and liquid parameters on the resonance frequency and scattering cross sections are analyzed.
Khismatullin et al. (2002) studied this question.