Attenuation and dispersion of sound in bubbly fluids via the Kramers—Kronig relations

Abstract

Sound propagation in a dilute bubble–liquid mixture is studied by means of the Kramers–Kronig relationships, which relate the real and imaginary parts of the general susceptibility of a linear medium. These relationships are adopted for the case of acoustic waves, where they become coupled integral equations. A simple but approximate procedure is used to obtain from these equations the phase speed of sound waves for the case when the attenuation coefficient is independently known. The procedure can be used to obtain the speed of propagation of sound waves in acoustic media having internal dissipation, but is here applied only to fluids containing radially pulsating bubbles. Approximate results for the speed of propagation and for the attenuation per wavelength are obtained for this case on the basis of a first-order estimate for the attenuation coefficient. These results are the same as those derived previously on the basis of model equations for bubbly liquids. They therefore provide additional support for those equations, while indicating some of their limitations.