A method for estimating time-dependent acoustic cross-sections of bubbles and bubble clouds prior to the steady state

Models for the acoustic cross-sections of gas bubbles undergoing steady-state pulsation in liquid have existed for some time. This article presents a theoretical scheme for estimating the cross-sections of single bubbles, and bubble clouds, from the start of insonation onward. In this period the presence of transients can significantly alter the cross-section from the steady-state value. The model combines numerical solutions of the Herring-Keller model with appropriate damping values to calculate the extinction cross-section of a bubble as a function of time in response to a continuous harmonic sound field (it is also shown how the model can be adapted to estimate the time-dependent scatter cross-section). The model is then extended to determine the extinction cross-section area of multiple bubbles of varying population distributions assuming no bubble-bubble interactions. The results have shown that the time taken to reach steady state is dependent on the closeness of the bubble to resonance, and on the driving pressure amplitude. In the response of the population as a whole, the time to reach steady state tends to decrease with increasing values of the driving pressure amplitude; and with the increasing values of the ratio of the numbers of bubbles having radii much larger than resonance to the number of resonant bubbles. The implications of these findings for the use of acoustic pulses are explored.