Acoustic intensity of narrowband sources using the phase and amplitude gradient estimator method

The phase and amplitude gradient estimator (PAGE) method [D.C. Thomas et al., J. Acoust. Soc. Am. 137, 3366-3376 (2015)] has proven successful in improving the accuracy of measured energy quantities over the traditional p-p method in several applications. One advantage of the PAGE method is the use of phase unwrapping, which can provide increased measurement bandwidth. However, narrowband sources often do not have coherent phase information over a sufficient bandwidth for a phase unwrapping algorithm to unwrap properly. Still, the PAGE method yields correct acoustic intensity measurements for frequencies up to the spatial Nyquist frequency for both broadband and narrowband sources. This is an improved bandwidth over the traditional method. For narrowband sources above the spatial Nyquist frequency, special conditions or processing are necessary for the PAGE method to provide accurate acoustic intensity. With sufficient bandwidth and a coherence of at least 0.1 at the spatial Nyquist frequency, a relatively narrowband source above the spatial Nyquist frequency can be unwrapped accurately. Also, with an assumption of a propagating field, and therefore linear phase, the extrapolated PAGE method uses the phase of a tone below the spatial Nyquist frequency to extrapolate phase above the spatial Nyquist frequency.The phase and amplitude gradient estimator (PAGE) method [D.C. Thomas et al., J. Acoust. Soc. Am. 137, 3366-3376 (2015)] has proven successful in improving the accuracy of measured energy quantities over the traditional p-p method in several applications. One advantage of the PAGE method is the use of phase unwrapping, which can provide increased measurement bandwidth. However, narrowband sources often do not have coherent phase information over a sufficient bandwidth for a phase unwrapping algorithm to unwrap properly. Still, the PAGE method yields correct acoustic intensity measurements for frequencies up to the spatial Nyquist frequency for both broadband and narrowband sources. This is an improved bandwidth over the traditional method. For narrowband sources above the spatial Nyquist frequency, special conditions or processing are necessary for the PAGE method to provide accurate acoustic intensity. With sufficient bandwidth and a coherence of at least 0.1 at the spatial Nyquist frequency, a relativel...