# Flavi Detection on a Tilted Particleboard by Use of the Spec-Radiation Method
## 2 Materials and Methods
### The Spec-Radiation Method for Tilted Particleboards  
With the spec-radiation method, it is possible, as mentioned in Section 2.1, to determine the sound distribution on a propagation plane parallel to the measuring plane. In order to achieve a higher repetition speed during the transmission of ultrasonic pulses and to avoid a standing wave, it is necessary to select a tilted arrangement. However, if the particleboard is now tilted to the measuring plane and the parallel plane is determined as usual at the distance of \(z=d\), it is possible that flaws that are not in the plane cannot be identified. The flaw could be in front or behind the calculated plane and thus escape identification. This is depicted schematically in Figure 2. It would now be possible to think about a simple coordinate transformation in the spatial domain \((x,y,z)\). However, this is not possible when working with Fourier transformations. Therefore, the coordinate transformation has to be conducted directly in the Fourier domain \((k_{x},k_{y},k_{z})\) because of the non-linear relation of the angular frequencies. This is described below and the derivation is mainly based on [35].  
Figure 1: Schematic representation of the spectrum on the Ewald sphere in the Fourier domain.  
Figure 2: Schematic representation of a tilted particleboard with flaws far away from the propagation plane.  
We start by considering that the plane of interest is at \(z=0\), with a coordinate transformation matrix \(\mathbf{T}\), which is as follows:  
\[\mathbf{T}=\begin{bmatrix}a_{1}&a_{4}&a_{7}\\ a_{2}&a_{5}&a_{8}\\ a_{3}&a_{6}&a_{9}\end{bmatrix}. \tag{12}\]  
The wave number vector in the parallel plane is as described in Section 2.1:  
\[\mathbf{k}=[k_{x},k_{y},k_{z}]=2\pi[f_{x},f_{y},f_{z}]. \tag{13}\]  
The wave number vector of the tilted coordinate system is given by:  
\[\mathbf{k_{t}}=[k_{x,t},k_{y,t},k_{z,t}]=2\pi[f_{x,t},f_{y,t},f_{z,t}]. \tag{14}\]  
The index t indicates the tilted coordinate system. By means of the coordinate transformation matrix \(\mathbf{T}\), the original wave vector can now be transformed into a tilted vector and vice versa:  
\[\mathbf{k}=\mathbf{T}\mathbf{k_{t}}, \tag{15}\]  
\[\mathbf{k_{t}}=\mathbf{T}^{-1}\mathbf{k}. \tag{16}\]  
As we are interested in transferring our spectrum from the parallel plane to the tilted plane, we continue to work with Equation (16) and receive:  
\[f_{x,t}=a_{1}f_{x}+a_{2}f_{y}+a_{3}f_{z}, \tag{17}\]  
\[f_{y,t}=a_{4}f_{x}+a_{5}f_{y}+a_{6}f_{z}. \tag{18}\]  
It is to be considered here that according to Equation (5):  
\[f_{z}=\sqrt{\left(\frac{1}{\lambda}\right)^{2}-f_{x}^{2}-f_{y}^{2}}, \tag{19}\]  
applies. This results in the spectrum:  
\[p(f_{x,t},f_{y,t},0,\omega)=p(a_{1}f_{x}+a_{2}f_{y}+a_{3}f_{z},a_{4}f_{x}+a_{5 }f_{y}+a_{6}f_{z},0,\omega). \tag{20}\]  
Due to the nonlinear coordinate transformation, the Jacobi determinant must be taken into account. From this, it follows that:  
\[\det(J)=\frac{\partial f_{x,t}}{\partial f_{x}}\frac{\partial f_{y,t}}{ \partial f_{y}}-\frac{\partial f_{x,t}}{\partial f_{y}}\frac{\partial f_{y,t }}{\partial f_{x}}. \tag{21}\]  
This leads to:  
\[\det(J)=\frac{(a_{1}a_{6}-a_{3}a_{5})f_{x}}{f_{z}}+\frac{(a_{3}a_{4}-a_{1}a_{ 6})f_{y}}{f_{z}}+(a_{1}a_{5}-a_{2}a_{4}). \tag{22}\]  
By applying an inverse Fourier transformation, the complex amplitudes of the tilted plane can be obtained by:  
\[p(x_{t},y_{t},0,\omega)=\mathfrak{F}^{-1}\left(p(f_{x,t},f_{y,t},0,\omega)\, \det(J)\right). \tag{23}\]However, there are still some peculiarities that have to be considered for the numerical implementation, which will be discussed in the following section.