# The Influence of Sensor Size on Acoustic Emission Waveforms--A Numerical Study
## 4 Discussion
### Impact of Sensors Size Effect on Frequency Content  
The sensor size effect does not only influence the amplitude but also the wave shape in both time and frequency domain. Considering the frequency content, the most crucial measure of the similarity between two waveforms (X(\(t\)), Y(\(t\))) is the \"coherence function, \(\gamma_{xy}\)\". This function is given by:  
\[\gamma_{xy}^{2}(f)=\frac{G_{xy}^{2}(f)}{G_{xx}(f)G_{yy}(f)},\hskip 28.452756 pt0 \leq\gamma_{xy}^{2}(f)\leq 1 \tag{2}\]  
where: and \(G_{xx}(f)\), \(G_{yy}(f)\) are the autospectral density functions of X(\(t\)) and Y(\(t\)) respectively and \(G_{xy}(f)\) is the cross-spectral density function between X(\(t\)) and Y(\(t\)). Coherence is analogous to the squared correlation coefficient of time domain functions underlying the frequency similarities of the signals [23]. In case of identical emitted and received waveforms, coherence gets a unity value. Coherence has also been measured to classify AE [24] and acousto-ultrasonics signals [25] based on their similarity to reference waveforms. In this case it was calculated based on a standard Matlab function. The waveforms were zero-padded to the length of 32,768 points while the required overlap window was 1250 points. These settings were constant for all calculations below but are not unique and could be changed leading to a somehow different final function. Figure 9(a) shows indicatively the wave emitted with excitation frequency at 1 MHz (in red), and the waveforms received by the medium sensor (5 mm) for the cases of source set on the surface (in black) and vertically beneath the sensor (in blue). It is shown that the similarity to the original wave is stronger for the vertically placed source, when two clear cycles are depicted, while for the case of surface waves, the major content of the Rayleigh waves shows four to five cycles instead of two. Figure 9(b) shows the corresponding coherence functions between the received waveforms and the excitation up to 2 MHz. The level of coherence is much higher for the direct propagation vertical to the sensor where the function is almost constantly close to unity, indicative of excellent similarity between the excitation and the sensor output. On the other hand, the coherence function of surface waves has significantly lower values with an average around 0.6 indicative of strong distortion that affects the received waveform. Figure 10: (**a**) Waveforms received by the 5 mm size sensor emitted from source set on the surface (in black) and vertically beneath the sensor (in blue) with excitation frequency equal to 1 MHz. Original waveform is added in red color and in reduced scale to fit the graph; (**b**) Respective coherence functions between received waves and original. Figure 9: Relative amplitude vs. sensor size over wavelength parameter (\(D/\lambda\)). The average level of coherence between the excitation and the received signal is shown in Figure 11 for the different angles of incidence and the various sensors sizes for two indicative excitation frequencies (a) 1 MHz and (b) 500 kHz. For the case of 1 MHz the coherence is quite high (more than 95%) for diagonal and vertical to the sensor surface propagation for all sensors. However, for propagation parallel to the surface (angle 0\({}^{\circ}\)), coherence is lower. Still, again the point sensor yields higher values (0.66) compared to the longer sensor (0.49). For the surface propagation of 500 kHz (case b), coherence is very low (below 0.3) showing a huge amount of distortion on the frequency content for all sensors, while it is again close to unity for diagonal and vertical direction of incidence. The above coherence analysis indicates that the received signal is crucially distorted in spectral content for surface propagation meaning that sources on or close to the surface will exhibit lower spectral similarity to the finally obtained waveform than a source at the same distance beneath the sensor.