A systematic method to obtain 3D finite-difference formulations for acoustic intensity and other energy quantities

A systematic method is described to obtain formulations based on the finite-difference approximation for computation of the energy quantities of three-dimensional (3D)-sound fields from measurements. It uses up to the first-order terms of the Taylor series expansion of the sound pressure about the microphone positions of a particular probe configuration. The use of a symbolic computation allows approximate expressions to be obtained for all energy quantities as a function of the cross-spectral densities of the microphone signals. The application of this approach is illustrated by the three components of the active acoustic intensity vector and the energy densities for different types of 3D-probes consisting of 4–6 microphones. The inherent effect associated with the finite-sum and finite-difference approximations is evaluated for each of these probes by considering 1D sound fields in order to make a comparison of their performance.