# The Spec-Radiation Method for Layered Fluid Media
## 3 Results
### Experimental Application  
For the evaluation, the measured area was increased symmetrically by zero-padding to a size of 256 \(\times\) 256 points with the measured area inside. With the spec-radiation method, the sound field on the board surface can be computed. The important frequency is the exciting frequency. Therefore, the evaluation was performed for a frequency of 50 kHz. The result of this evaluation is depicted in Figure 10. In both results, the flaw may be detected, but not accurate in shape and size. This was due to the fact that the sound field behind the flaw had diffraction and interference patterns that affected the sound field on the board surface. In (a) is the result of the 10 mm flaw and in (b) the result of the 6 mm flaw. In the next step, we evaluated the result with the proposed extension, so that the result was evaluated at the \(z\)-position of the flaw in the particleboard. Figure 9: Maximum sound pressure in dB related to the maximum in the entire volume. Perfect impedance matching of a 3-layer model. The results were evaluated for frequencies up to 250 MHz. We therefore calculated this through the particleboard. In contrast to the calculation of the forward propagation, it has to be considered that the start plane was the measuring plane. This means that the start sound pressure was the transmitted sound pressure \(P_{\text{T}}(f_{x},f_{y},z,\omega_{\text{j}})\). To obtain the incidence sound pressure at the interface, the transmitted sound pressure has to be divided by \(T_{\delta}\), so that:  
\[P(f_{x},f_{y},z,\omega_{\text{j}})=\frac{1}{T_{\delta}}P_{\text{T}}(f_{x},f_{ y},z,\omega_{\text{j}}). \tag{25}\]  
The result is depicted in Figure 11. In (a), the 10 mm flaw is clearly visible as a circle with a diameter of 10 mm and the pressure drops stronger than \(-20\) dB. In (b), the flaw with a diameter of 6 mm is visible more clearly, but is still not perfect. The area is larger than a circle of 6 mm. This was due to the fact that the 6 mm circle was close to the detection limit of \(\lambda/2\) which was, in the wooden particleboard, approximately 4.5 mm. The pressure drop was not as strong, so that some air between the tape and the particleboard around the flaw could have raised the area here. The flaw was easier to detect in comparison with the result at the top surface of the particleboard. The evaluation time took approximately 14 s per interface layer with the corresponding dataset. Figure 11: Evaluation result on the lower surface of the wooden particleboard, with (a) 10 mm and (b) 6 mm under the particleboard. The results were evaluated at a frequency of 50 kHz. Figure 10: Evaluation result on the top surface of the wooden particleboard, with (a) 10 mm and (b) 6 mm under the particleboard. The results were evaluated at a frequency of 50 kHz. By calculating many planes from the microphone plane to the flaw plane, the sound field can also be represented as a cross-section at \(x=0\) m. This is depicted in Figure 12 for the \(\omega 10\) mm flaw. Here, two different scalings of the results were chosen. Everything above the particleboard (\(z=0\) m to \(-0.177\) m) was scaled in dB related to the maximum in the air volume above the particleboard. Everything inside the particleboard (\(z=-0.178\) m to \(-0.202\) m) was scaled in dB with respect to the maximum inside the particleboard. This resulted in a matching color scaling of the two areas. If everything would be scaled to the maximum inside the particleboard, it would no longer be possible to display the entire sound field in a well-recognizable color, because the sound field above the board would then be at approximately \(-55\) dB. From this cross-sectional view, it is clear that on the surface of the particleboard (\(z=-0.177\) m), the flaw could not be clearly detected. What could only be detected there was a minimum of the sound pressure due to the diffraction of the sound around the flaw, as can also be seen in Figure 10a. The flaw itself, however, was completely masked by the diffraction.