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More information for "Measurement" section
Sound intensity is defined as the time averaged product of sound pressure and acoustic particle velocity. Both quantities can be directly measured by using a sound intensity p-u probe comprising a microphone and a particle velocity sensor, or estimated indirectly by using a p-p probe that approximates the particle velocity by integrating the pressure gradient between two closely spaced microphones.

Pressure-based measurement methods are widely used in anechoic conditions for noise quantification purposes. The bias error introduced by a p-p probe can be approximated by $$\widehat{I}^{p-p}_n \simeq I_n - \frac{\varphi_{\text{pe}}\,p_{\text{rms}}^2}=I_n \biggl( 1-\frac{\varphi_{\text{pe}}}\frac{p_{\text{rms}}^2/ \rho c}{I_r}\biggr) \, ,$$ where $$I_n$$is the “true” intensity (unaffected by calibration errors), $$\hat{I}^{p-p}_n$$ is the biased estimate obtained using a p-p probe, $$p_{\text{rms}}$$is the root-mean-squared value of the sound pressure, $$k$$ is the wave number, $$\rho$$ is the density of air, $$c$$ is the speed of sound and $$\Delta r$$ is the spacing between the two microphones. This expression shows that phase calibration errors are inversely proportional to frequency and microphone spacing and directly proportional to the ratio of the mean square sound pressure to the sound intensity. If the pressure-to-intensity ratio is large then even a small phase mismatch will lead to significant bias errors. In practice, sound intensity measurements cannot be performed accurately when the pressure-intensity index is high, which limits the use of p-p intensity probes in environments with high levels of background noise or reflections. On the other hand, the bias error introduced by a p-u probe can be approximated by $$\hat{I}^{p-u}_n=\frac{1}{2} \text{Re}\{{P\hat{V}^*_n}\}=\frac{1}{2} \text{Re}\{{P V^*_n \text{e}^{-\text{j}\varphi_{\text{ue}}} }\} \simeq I_n + \varphi_{\text{ue}} J_n  \, ,$$ where $$\hat{I}^{p-u}_n$$ is the biased estimate obtained using a p-u probe, $$P$$ and $$V_n$$ are the Fourier transform of sound pressure and particle velocity, $$J_n $$is the reactive intensity and $$\varphi_{\text{ue}} $$is the p-u phase mismatch introduced by calibration errors. Therefore, the phase calibration is critical when measurements are carried out under near field conditions, but not so relevant if the measurements are performed out in the far field. The “reactivity” (the ratio of the reactive to the active intensity) indicates whether this source of error is of concern or not. Compared to pressure-based probes, p-u intensity probes are unaffected by the pressure-to-intensity index, enabling the estimation of propagating acoustic energy in unfavorable testing environments provided that the distance to the sound source is sufficient.