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