# Design of a Phase Calibrator to Evaluate Non-Matched Microphones for Intensity Probes
## 2 Materials and Methods
### Numerical Model Definition
To summarize, the model characteristics were the following:  
* Cylindrical and rectangular geometry as reported in Figures 1 and 2;
* Simplified geometry of the acoustic volume;* Solid CTETRA fluid nodes elements;
* Rigid wall boundary condition;
* Imposed pressure on constrained nodes to model the acoustic load;
* SOL108--Direct Frequency Response Analysis. By way of illustration, we have provided a detailed analysis of the response behavior of the acoustic volume in terms of amplitude and phase differences as evaluated in correspondence with the fluid nodes that model the microphones in the [1000-3500] Hz frequency range. We have, therefore, decided to provide the frequency response functions as averaged on the nine points (one central point and eight surrounding points) that meshed the acoustic volume indentation representing the microphones themselves. The cylindrical configuration was the one of choice for this operation. Figures 3-5 show the amplitude, the amplitude ratio and phase of the frequency response function associated with the cylindrical configuration in the [1000-3500] Hz frequency range. It is possible to notice an occurrence of possible acoustic volume mode shapes around 1500 Hz and 2500 Hz. This eventuality can also be emphasized in the experimental results for which similar behavior can be found around 1500 Hz and 2500 Hz. The deviation of the experimental results from the numerical simulation could be due to non-linearities that were not included in the model and other factors such as local roughness of the material and non perfectly rigid wall of the physical calibrator itself. It is interesting to notice that even at probable resonances of the volume, the microphone's response is almost overlapped. Figure 1: Cylindrical calibrator FEM model. Figure 2: Rectangular calibrator FEM model. As from the previous graphs, those differences are very small (if compared to the total value of the acoustic field measured) in terms of amplitude and are numerically neglectable in terms of phases. This result was in some way expected to be the acoustic wavelength  
Figure 4: Cylindrical calibrator FRF amplitude ratio [1000–3500] Hz. Figure 3: Cylindrical calibrator FRF [1000–3500] Hz—Amplitude. (\(\lambda=C/f\) where \(C\) is the acoustic wave speed in air and \(f\) is the frequency) bigger than the microphone's membrane diameter in the investigated frequency range. Let us notice that, as reported in Figures 6-9, the microphones are exposed to the same acoustic field in the over-mentioned frequency range. As a final result of the numerical investigation, it resulted that within the frequency range considered, both cylindrical and parallelepiped calibrators' shapes had a very low effect on the acoustic differences expected on the two microphone locations, and, as a consequence, both shapes were produced and tested. Figure 5: Cylindrical calibrator FRF [1000–3500] Hz—Phase. \[\frac{\partial^{2}p}{\partial^{2}x}+\frac{\partial^{2}p}{\partial ^{2}y}+\frac{\partial^{2}p}{\partial^{2}z}=\frac{1}{c^{2}}\frac{\partial^{2}p} {\partial^{2}t}. \tag{6}\]0 Cylindrical calibrators’ pressure ISO surface for microphone exposition evaluation— general view.