## Materials and Methods
### Mathematical Methods  
#### 2.2.1 Re-Radiation  
The re-radiation method is based on the Rayleigh-Sommerfeld diffraction integral. This is based on the wave equation:  
\[\bigtriangledown^{2}p-\frac{1}{c^{2}}\frac{\partial^{2}p}{\partial t^{2}}=0. \tag{1}\]  
Figure 3: Experimental setup with a flaw imitation at the top of the wooden particleboard. NI = National Instruments.  
Figure 2: Schematic experimental setup with a flaw imitation at the top of the wooden particleboard.  
where \(p\) is the acoustic pressure, \(c\) the speed of sound, \(t\) the time, and \(\bigtriangledown\) is the nabla-operator. [PERSON] has already shown [23] how the diffraction integral is obtained by applying [PERSON]'s theorem from the wave equation. This correlation from optics can also be applied to the acoustics of gaseous media and has already been used in in previous works [16; 17; 18; 19; 20; 22], as well as in NDT, and is only briefly presented and explained here. The pressure value in the propagation plane (Figure 4) is the integral of the pressure value in the measuring plane multiplied by the derivative of the Green's function. It can be understood as the sum of the respective parts of the measuring plane for the wave propagation to the propagation plane. The [PERSON] diffraction integral in the frequency domain in its unrestricted form is:  
\[p(x,y,z,\omega)=2\int_{S}p(x^{\prime},y^{\prime},0,\omega)\frac{\partial G}{ \partial z}\mathrm{d}x^{\prime}\mathrm{d}y^{\prime}. \tag{2}\]  
It describes the integral of the acoustic pressure over an entire known area S multiplied by the derivative of the free space Green's function. \(x\), \(y\), \(z\) are Cartesian coordinates. \(G\) is the free space Green's function [23]:  
\[G=-\frac{\mathrm{e}^{(\mathrm{Vik}\sqrt{(x-x^{\prime})^{2}+(y-y^{\prime})^{2 }+z^{2}})}}{4\pi\sqrt{(x-x^{\prime})^{2}+(y-y^{\prime})^{2}+z^{2}}}=-\frac{ \mathrm{e}^{\mathrm{Vik}R}}{4\pi R}. \tag{3}\]  
In this equation, i is the complex unit, \(k\) is the wave number, and \(V\) is the propagation parameter and for the back propagation \(-1\). R is the direct distance of one point in the known plane to one point in the propagation plane. The derivative of the free space Green's function for the backward propagation is:  
\[\frac{\partial G}{\partial z}=\frac{\mathrm{ze}^{-\mathrm{i}kR}}{4\pi R^{2}} \left(\frac{1}{R}+\mathrm{i}k\right). \tag{4}\]  
In order to solve the Rayleigh-Sommerfeld diffraction integral for practical applications, [PERSON] [24] has suggested to performing the convolution in the wave number space because this is a very time efficient way to compute the integral numerically. Thus, the equation used here for the re-radiation method results in  
\[p(x,y,z,\omega)\cong 2h^{2}p(x^{\prime},y^{\prime},0,\omega)*\frac{\mathrm{ze}^{ -\mathrm{i}kR}}{4\pi R^{2}}\left(\frac{1}{R}+\mathrm{i}k\right). \tag{5}\]  
The asterisk is the convolution operator, and \(h^{2}\) is the area around each point in the plane, as depicted in Figure 4.  
Figure 5a depicts an airborne sound field that was calculated utilizing the re-radiation method. On the right side is the measuring plane, and the left side shows the surface of a particleboard that was excited with the transmitter; at the surface of the particleboard, there is a 20 mm flaw. The propagation  
Figure 4: Schematic representation of the re-radiation method.  
plain is always the air layer directly above of the particleboard, like that shown in Figure 5b. All results of the re-radiation method are further depicted for a frequency range from 48 to 52 kHz.