# Design of a Phase Calibrator to Evaluate Non-Matched Microphones for Intensity Probes
## 1 Introduction  
Sound intensity measurements represent a standard for determining the acoustic power that a piece of generic machinery can radiate [1; 2; 3]. Its success is based on the peculiar characteristic of not being influenced by the surrounding acoustic environment. This has been obtained, until today, by building acoustic intensity probes only by \"matched\" microphones guaranteeing the maximum correlation between their measurements in terms of amplitude and phase responses. This correlation is also periodically verified by very expensive calibrators, in which the two microphones are exposed contemporarily at the same acoustic fields, providing frequency response functions in which the amplitude represents the ratio of acoustic magnitudes and the difference between the two respective microphones phases (due only to intrinsic mismatch). Sound intensity probes are, therefore, complex devices that allow accurate measurements but are not always sustainable for companies or professionals. Therefore, as the first step toward the development of a more cost-effective instrument, we have designed an acoustic calibrator that can match the phases of two generic microphones not explicitly designed for intensity probes. The sound intensity is a measurement of the acoustic energy that flows through a unit area along a direction perpendicular to it. The intensity can be expressed through the product between the pressure (\(p\)) and the particle velocity component (\(v\)), as represented in Equation (1).  
\[I=pv \tag{1}\]The average sound intensity during time \(T\) is given by Equation (2):  
\[<I>=\frac{1}{T}\int_{0}^{T}p(t)u(t)dt \tag{2}\]  
In the case of the so-called \(p\)-\(p\) probes, the intensity measurement occurs by deploying two closely placed microphones, appropriately spaced concerning an imposed frequency error bound [4; 5]. From a mathematical formulation, the particle velocity can be obtained with the linearized Euler equation; it is possible to assume the pressure gradient as the pressure difference on the distance between the two microphones [6; 7; 8].  
\[u(t)_{r}=-\int_{-\infty}^{t}\frac{p(t)_{2}-p(t)_{1}}{\Delta r\rho} \tag{3}\]  
From Equation (3), in which \(p_{1,2}\) is the pressure as measured by the microphones while \(\Delta r\) and \(\rho\) represent, respectively, the fixed distance between the microphones and the fluid density, it is evident that the particle velocity can be linked to the pressure difference between the microphones and, therefore, to the time delay between the signals. That translates into a frequency mismatch between the signals. By considering the formulation reported in [9], it is possible to compute the sound intensity by evaluating the imaginary part of the cross-spectrum \(G(f)_{1,2}\), which represents the phase difference itself as a function of the frequency \(f\). This formulation is reliable if it is possible to state that the pressure gradient between the microphones is small. This condition is reported in Equation (4) in which \(k\) represents the wave number.  
\[\begin{split} I(f)=\frac{[G(f)_{1,2}]}{2\pi f\rho\Delta r}\\ k\Delta r<<1\end{split} \tag{4}\]  
Given the theoretical background, it is clear that before proceeding with the sound intensity measurement, it is necessary to address any intrinsic phase mismatch between the microphones [10; 11; 12]. With \"intrinsic\", we refer to the phase mismatch that the two microphones carry as a bias error and therefore is not linked to the pressure difference generated by the measurement itself. Current calibration methods are highlighted in [13; 14; 15; 16]. This paper aims at designing a cost-effective phase calibrator to evaluate the phase mismatch between two generic class-1 microphones, not intended for intensity measurements.