Energy quantities, which are calculated from pressure and particle velocity, yield a great deal of information about acoustic fields. Errors in pressure or particle velocity estimation lead to bias errors the estimation of energy quantities. The bias errors arise from different probe configurations and processing methods. Two processing methods are examined: the traditional method and the recently developed Phase and Amplitude Gradient Estimation (PAGE) method. These two methods are compared to investigate how each estimates pressure and particle velocity and the subsequent bias errors in a plane wave, standing wave, and spherical spreading wave field. Analytical expressions are derived for the energy quantity estimation using ideal one-dimensional probes. A simulation of the field from a baffled circular piston and measurements using ideal two-dimensional probes is computed. Compared to the traditional method, the PAGE method significantly extends the range of frequencies for which the results are accurate. It is found that a probe with a center microphone significantly reduces the estimation error and extends the usable range of frequencies. The PAGE method with unwrapping, perfectly matches the analytical results for plane waves, while the traditional method is only good at wavelengths that are large compared to the probe size. Furthermore, the PAGE method has a constant bias error in spherical wave fields due to the 1/r decrease in pressure. The traditional method has a frequency dependent bias error that is much worse at higher frequencies. Lastly, the PAGE method has the same or worse error for the standing wave. As an application of energy quantities, acoustic intensity is used to develop an equivalent source model for jet noise from an F-22 at military and afterburner engine conditions. An optimization is used to find the best-matching wavepacket model for measured intensity vectors. The results are compared to another intensity method of estimating the source region and source directivity, and the two methods have good agreement.

Energy-based acoustic quantities provide vital information about acoustic fields and the characterization of acoustic sources. Recently, the phase and amplitude gradient estimator (PAGE) method has been developed to reduce error and extend bandwidth of energy-based quantity estimates. To inform uses and applications of the method, analytical and experimental characterizations of the method are presented. Analytical PAGE method bias errors are compared with those of traditional estimation for two- and three-microphone one-dimensional probes. For a monopole field when phase unwrapping is possible, zero bias error is achieved for active intensity using three-microphone PAGE and for specific acoustic impedance using two-microphone PAGE. A method for higher-order estimation in reactive fields is developed, and it is shown that a higher-order traditional method outperforms higher-order PAGE for reactive intensity in a standing wave field. Extending the applications of PAGE, the unwrapped phase gradient is used to develop a method for directional sensing with improved bandwidth and arbitrary array response.