The scattering of plane, transient acoustic waves by soft bodies is examined from the standpoint of shock-wave attenuation in fluids and solids. Asymptotic results in powers of scatterer stiffness are obtained that establish a stiffness range for which substantial and thorough attenuation is possible. This stiffness range lies between two extremes: one in which the scatterer acts as a rigid inclusion offering no pressure reduction benefits, and the other in which the scatterer suffers ultimate collapse in the manner of a cavity, with the concomitant generation of a large rebound pulse.

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