ACOUSTIC PROPAGATION THROUGH BUBBLES: AN EXPLORATION OF THE 1^ST AND 2^ND MOMENTS IN VARIOUS FLOW CONDITIONS

The classic theory for the propagation of acoustic waves in bubbly fluids, which is the 1^st moment of the multiple scattering solution, is well known. An integral equation describing the 2^nd moment can also be formulated using the multiple scattering approach, although solutions to this equation are difficult and typically require approximations. In this paper, the multiple scattering approach is used to derive a solution to the 2^nd moment in a slightly different fashion. This is done by first considering the solution to the 2^nd moment for the single scatterers only, following with the solution for the double scatterers, and eventually generalizing to the complete solution. The solutions to both the 1^st and 2^nd moments require no knowledge of any flow parameters, although they do require averaging over all bubble configurations. The time required for enough bubble configurations to be observed forms the basis for an indirect link between the multiple scattering theory and the flow. This relationship between the statistics of the propagation through bubbles and the flow in which the bubbles exist is explored in a laboratory environment. Multi-frequency attenuation measurements are inverted to estimate the bubble population distribution, which in turn is used to predict the 2^nd moment. The predicted 2^nd moment is then compared to the measured 2^nd moment. The time scales associated with the convergence of the measurements to the prediction is then estimated for the flow conditions investigated, providing an estimate of the mixing time of the flow. Although these time scales are specific to the laboratory studies described herein, it is hoped that they will serve to help illustrate the relationship between the multiple scattering theory of bubbles and the flow in which they reside.