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Calculation of acoustic intensity using the phase and amplitude gradient estimator (PAGE) method has been shown to increase the effective upper frequency limit beyond the traditional p-p method when the source of interest is broadband in frequency [Torrie, Whiting, Gee, Neilsen, and Sommerfeldt, Proc. Mtgs. Acoust. 23, 030005 (2015)]. PAGE processing calculates intensity for narrowband sources without bias error up to the spatial Nyquist frequency [Succo, Sommerfeldt, Gee, and Neilsen, Proc. Mtgs. Acoust. 30, 030015 (2018)]. The present work demonstrates that for narrowband sources with frequency content above the spatial Nyquist frequency, additive low-level broadband noise can improve intensity calculations. To be effective, the angular separation between the source and additive noise source should be less than 30 ° , while using phase unwrapping with a smaller angular separation will increase the usable bandwidth. The upper frequency limit for the bandwidth extension depends on angular separation, sound speed, and probe microphone spacing. Assuming the signal-to-additive-noise ratio ( SNR a ) is larger than 10 dB, the maximum level and angular bias errors incurred by the additive broadband noise beneath the frequency limit—or up until probe scattering effects must be taken into account—are less than 0.5 dB and 2.5 ° , respectively. Smaller angular separation yields smaller bias errors. [ABSTRACT FROM AUTHOR]

Bias errors for two-dimensional active acoustic intensity using multi-microphone probes have been previously calculated for both the traditional cross-spectral and the Phase and Amplitude Gradient Estimator (PAGE) methods [Whiting, Lawrence, Gee, Neilsen, and Sommerfeldt, J. Acoust. Soc. Am. 142, 2208–2218 (2017)]. Here, these calculations are expanded to include errors due to contaminating noise, as well as probe orientation. The noise can either be uncorrelated at each microphone location or self-correlated; the self-correlated noise is modeled as a plane-wave with a varying angle of incidence. The intensity errors in both magnitude and direction are dependent on the signal-to-noise ratio (SNR), frequency, source properties, incidence angles, probe configuration, and processing method. The PAGE method is generally found to give more accurate results, especially in direction; however, uncorrelated noise with a low SNR (below 10–15 dB) and low frequency (wavelengths more than 1/4 the microphone spacing) can yield larger errors in magnitude than the traditional method—though a correction for this is possible. Additionally, contaminating noise does not necessarily impact the possibility of using the PAGE method for broadband signals beyond a probe's spatial Nyquist frequency. [ABSTRACT FROM AUTHOR]

In acoustic intensity estimation, adding a microphone at the probe center removes errors associated with pressure averaging. Analytical bias errors are presented for a one-dimensional, three-microphone probe for active intensity, reactive intensity, and specific acoustic impedance in a monopole field. Traditional estimation is compared with the Phase and Amplitude Gradient Estimator (PAGE) method; the PAGE method shows an increased bandwidth for all three quantities. The two- and three-microphone methods are compared experimentally, showing reduced bias errors with three-microphone PAGE for active and reactive intensity, whereas using two microphones is preferred for specific acoustic impedance. [ABSTRACT FROM AUTHOR]

Sound intensity measurements using two microphones have traditionally been processed using a cross-spectral method with inherent error in the finite-sum and finite-difference formulas. The phase and amplitude gradient estimator method (PAGE) has been seen experimentally to extend the bandwidth of broadband active intensity estimates by an order of magnitude. To provide an analytical foundation for the method, bias errors in active intensity and specific acoustic impedance are presented and compared to those of the traditional method. Bias errors are reported for a planewave field and sound radiated from a monopole and a dipole. Additionally, bias errors are reported for reactive intensity, the estimation of which is unchanged by the PAGE method for the two-microphone case. [ABSTRACT FROM AUTHOR]

The phase and amplitude gradient estimator (PAGE) method for active acoustic intensity uses pairwise microphone transfer functions to obtain the phase gradient, which improves the calculation bandwidth over the traditional weighted quadspectral method. Additionally, for broadband sources, the PAGE theory indicates that the transfer function phase can be unwrapped to further extend the usable frequency range to beyond the spatial Nyquist frequency. Here, two experiments demonstrate intensity bandwidth extension by more than an order of magnitude using phase unwrapping. First, plane-wave tube results show accurate one-dimensional intensity calculations with the microphones separated by five wavelengths, 30 times the traditional limit. Second, two-dimensional measurements of a laboratory-scale jet with a four-microphone probe yield physically reasonable calculations at frequencies 15 times the traditional limit. [ABSTRACT FROM AUTHOR]