# Design of a Phase Calibrator to Evaluate Non-Matched Microphones for Intensity Probes
## 2 Materials and Methods
### Numerical Model Definition  
In order to design a calibrator that allowed the two microphones to be exposed to the same acoustic field, numerical simulations were carried out as frequency responses until about 3500 Hz, comparing the differences in terms of amplitude and phase expected at microphone locations. Those are the minimum differences we can expect to measure in a nominally perfect calibrator (with no geometry tolerances due to production process) characterized by those geometries and measured by two perfectly matched microphones. Numerical acoustical FEM models of the calibrator configurations were implemented using commercial software originally intended for structural applications [17]. This approach relies upon the so-called structural-acoustic analogy, which provides the possibility of relating structural displacements to acoustic pressure [18]. Let us consider the standard scalar wave Equation (5), which can be written in the Cartesian coordinates Equation (6):  
\[\
abla^{2}p=\frac{1}{c^{2}}\frac{\partial^{2}p}{\partial^{2}t} \tag{5}\] \[\frac{\partial^{2}p}{\partial^{2}x}+\frac{\partial^{2}p}{\partial ^{2}y}+\frac{\partial^{2}p}{\partial^{2}z}=\frac{1}{c^{2}}\frac{\partial^{2}p} {\partial^{2}t}. \tag{6}\]  
Equation (6) can be easily reconducted to Equation (7), which represents the stress equilibrium in materials for a given direction. \[\frac{\partial\sigma_{xx}}{\partial x}+\frac{\partial\tau_{xy}}{\partial y}+ \frac{\partial\tau_{xx}}{\partial z}=\rho_{s}\frac{\partial^{2}u_{x}}{ \partial^{2}t} \tag{7}\]  
Equations (6) and (7) are similar if we consider that:  
\[\sigma_{xx}=\frac{\partial p}{\partial x};\tau_{xy}=\frac{\partial p}{ \partial y};\tau_{xx}=\frac{\partial p}{\partial z};\rho_{s}=\frac{1}{c^{2}}; u_{x}=p. \tag{8}\]  
Therefore, by applying the structural-acoustic analogy, it is possible to deploy an FEM formulation that is described, for the acoustic case, by Equation (9), in which \([M_{a}]\) and \([K_{s}]\) represent, respectively, an acoustic mass and stiffness matrix, while \(F\) represents the excitation force on the system. \[[M_{a}]\ddot{p}+[K_{s}]p=F \tag{9}\]  
To build the matrices in Equation (9), it is necessary to define the physical properties of the fluid nodes of the model [19]. The fluid volume has been meshed with CTETRA elements, whose size was chosen according to the model's required frequency range, which is spaced from [500-3500] Hz. CTETRA elements (3D solid elements) require a PSOLID property (primary property entry) that refers to a MAT110 card that defines the bulk modulus and other properties of the fluid such as density and speed of sound. Generally speaking, the acoustic field inside of the calibrator, generated from an excitation produced by a loudspeaker, could be modeled as an acousto-structural problem in which the motion of the membrane excites the calibrator's internal volume. In this specific case, to simplify the calculation, the membrane motion has been excluded from the model and the external acoustic load has been applied directly to constrained fluid nodes of the regions in which the speaker is supposed to be positioned. Let us notice that the exciting force has been considered unitary and constant throughout the frequency spectrum. The applied method requires also boundary conditions. In this case, the solid wall condition (no flow occurs across the boundary; pressure gradient equal to zero) appeared to be coherent with the phenomenon. The solution of choice was the SOL108, which implements a direct frequency response analysis. Let us notice that the microphones were modeled as fluid volume indentations on the face opposing the loudspeaker, with nodes for phase and amplitude evaluation applied in the center part of the indentation itself and on eight surrounding points.