Near resonant bubble acoustic cross-section corrections, including examples from oceanography, volcanology, and biomedical ultrasound

Abstract

The scattering cross-section sigma(s) of a gas bubble of equilibrium radius R(0) in liquid can be written in the form sigma(s)=4piR(0) (2)[(omega(1) (2)omega(2)-1)(2)+delta(2)], where omega is the excitation frequency, omega(1) is the resonance frequency, and delta is a frequency-dependent dimensionless damping coefficient. A persistent discrepancy in the frequency dependence of the contribution to delta from radiation damping, denoted delta(rad), is identified and resolved, as follows. Wildt's [Physics of Sound in the Sea (Washington, DC, 1946), Chap. 28] pioneering derivation predicts a linear dependence of delta(rad) on frequency, a result which Medwin [Ultrasonics 15, 7-13 (1977)] reproduces using a different method. Weston [Underwater Acoustics, NATO Advanced Study Institute Series Vol. II, 55-88 (1967)], using ostensibly the same method as Wildt, predicts the opposite relationship, i.e., that delta(rad) is inversely proportional to frequency. Weston's version of the derivation of the scattering cross-section is shown here to be the correct one, thus resolving the discrepancy. Further, a correction to Weston's model is derived that amounts to a shift in the resonance frequency. A new, corrected, expression for the extinction cross-section is also derived. The magnitudes of the corrections are illustrated using examples from oceanography, volcanology, planetary acoustics, neutron spallation, and biomedical ultrasound. The corrections become significant when the bulk modulus of the gas is not negligible relative to that of the surrounding liquid.