## 2 Materials and Methods  
### Direct Synchronization Method--Principle of Operation  
The method with direct synchronization can be used when the sound source generating the analyzed sound field is an electroacoustic transducer. In such a situation, it is possible, using an appropriate measurement interface, to synchronize the process of generation of the signal applied to the transducer with the start of measurement data acquisition from the microphone placed in the first position. This requires the use of a digital-to-analogue converter synchronized by a common clock signal in the excitation signal generation system and an analogue-to-digital converter in the measurement data acquisition system. Such requirements are met by virtually every good quality audio interface today. Then, after the data acquisition process is completed, the microphone is placed in the second position and the measurement process is repeated (again, the excitation signal controlling the speaker is synchronized with the data acquisition). This will ensure that the recorded data files for successive microphone positions are synchronized to within one sampling period.  
Under the right conditions (primarily low noise floor level), ignoring external factors that disturb the sound field (stable experiment conditions), it can be simplified to assume that the generated sound field is identical in both measurements. In addition, thanks to synchronization, the acquisition is made in both measurements, at the identical phase of the signal generated by the transducer, and the phase shift between the two recorded signals, is the result only of a change in the position of the microphone. This corresponds to simultaneous measurement by two microphones in a classical PP probe. Of course, this assumes that the measurement conditions are unchanged.  
The instantaneous value of sound intensity is by definition the product of pressure and acoustic velocity. In a pressure-pressure (PP) intensity probe, the determination of the acoustic velocity is conducted indirectly. For this purpose, the linearized Euler equation is used, from which the relation for the acoustic velocity component in the x-direction is derived:  
\[v_{x}(t)=-\frac{1}{\rho_{0}}\int_{-\infty}^{t}\frac{\partial p(t)}{\partial x }d\tau \tag{1}\]  
where \(\rho_{0}\) is the air density.  
A PP-type probe uses a pair of microphones located at a small distance \(\Delta x\) from each other, and the derivative in Equation (1) is replaced by the differential quotient, yielding the relation:  
\[v_{x}(t)=-\frac{1}{\rho_{0}\Delta x}\int_{-\infty}^{t}(p_{1}(\tau)-p_{2}(\tau ))d\tau \tag{2}\]  
where \(p_{1}\) and \(p_{2}\) represent the pressures recorded by the two microphones of the PP probe. The sound pressure value used to calculate the sound intensity is determined as the average of the pressures \(p_{1}\) and \(p_{2}\):  
\[p=\frac{p_{1}+p_{2}}{2} \tag{3}\]  
Finally, each of the three orthogonal components of the sound intensity vector can be determined by measuring the two pressures measured by the microphones included in the PP probe aligned on a given axis according to Formula (4) [55]:  
\[I_{i}(t)\approx-\frac{1}{2\rho_{0}\Delta x}(p_{1}(t)+p_{2}(t))\int_{-\infty}^{ t}(p_{1}(\tau)-p_{2}(\tau))d\tau \tag{4}\]  
where \(\rho_{0}\) is the air density, \(\Delta x\) is microphone spacing, and \(p_{1}(t)\) and \(p_{2}(t)\) are pressure signals from first and second microphone position, respectively.  
The measurement process for the direct synchronization method is shown symbolically in Figure 1a,b, while Figure 1c presents tone signals explaining how the method works. The time \(\Delta t\) seen in Figure 1c was calculated using value of speed of sound in air at a given temperature.