Weakly nonlinear propagation of plane pressure waves in initially quiescent liquids containing spherical bubbles with a small initial polydispersity of the bubble radius and bubble number density is theoretically studied to clarify the effect of the polydispersity on wave propagation. The wavelength is comparable with the bubble radius and the incident frequency of waves is with an eigenfrequency of the bubble. We consider two different cases of the size of the polydispersity (i.e., small and large polydispersities) and assume that the polydispersity appears at a field far from a sound source. The polydispersity is formulated into a set of perturbation expansions. Bubble oscillations are spherically symmetric, and bubbles do not coalesce, break up, appear, and disappear. The basic set is composed of the conservation laws of mass and momentum for gas and liquid phases based on a two-fluid model, equation of motion for bubble wall, and so on. From the perturbation analysis up to the third order of approximation, two cases of nonlinear wave equation for a long-range propagation of waves were derived. We concluded that the polydispersity affects the advection of waves and is expressed as a variable coefficient.
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