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The oscillations of ultrasound contrast agents are of particular importanceto the understanding of the propagation of acoustic waves in the bubbly liquids(suspensions of ultrasound contrast agents). Acoustic waves propagating inbubbly liquids have been investigated extensively. Little has been dedicated tothe resonance effects of the microbubbles on the propagating waves. Here anonlinear partial differential equation for describing one-dimensional acousticwaves propagating near the resonance frequency of the microbubbles in bubblyliquids is obtained. The present equation recovers classical results forpropagating acoustic waves with finite amplitudes in liquids and interprets theacoustic localization in bubbly liquids explicitly.

Propagation of Nonlinear Acoustic Waves in the Suspension of Ultrasound  Contrast Agents Part I: Equation for Counting Resonance Effects and Revealing  Acoustic Localization