A three-dimensional finite-element analysis is used to determine the acoustic behavior of plane waves propagating through bubbly liquids having periodic bubble distributions. The results obtained from the model include the effective phase speed and attenuation of the medium, the acoustic field in the region of each bubble, and the virtual mass for both the monopole and dipole modes of bubble oscillation. The results are in agreement with both experimental data and previous analytical results.
The method of exhaustion is generalized to a simple formula that can be used to integrate functions under very general conditions, provided that the integral exists. Both a geometric proof (following the usual procedure for the method of exhaustion) and an independent algebraic proof are provided. Applications provided as examples include use of the formula to generate new series for common functions as well as computing the group velocity distribution resulting from waves diffracted from an aperture.
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