The propagation of ultrasonic waves through a bubbly liquid into tissue: a linear analysis

The steady-state response induced by a harmonically driven, ultrasonic wave in a structure comprised of two layers, the first a bubbly liquid, and the second a viscoelastic solid with a rigid boundary, is studied in the linear approximation. This structure is intended to model a cavitating liquid in contact with tissue. The upper surface of the liquid is driven harmonically and models the source. The lower surface of the solid is rigid and models the bone. While cavitation is inherently nonlinear, the propagation process is approximated as linear. The transient response is not calculated. The model of the bubbly liquid is a simple continuum one, supplemented by allowing for a distribution of different equilibrium bubble radii and for the relaxation of the oscillations of each bubble. The model contains three functions, the probability distribution describing the distribution of bubble radii, and two functions modeling the mechanical response of the individual bubble and the tissue, respectively. Numerical examples are worked out by adapting data taken from various published sources to deduce the parameters of these functions. These examples permit an assessment of the overall attenuation of the structure, and of the magnitudes of the pressure and particle velocity in the bubbly liquid and of the traction and the particle displacement in the tissue.<
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