# Deep Machine Learning for Path Length Characterization Using Acoustic Diffraction
## 1 Introduction
### Theory
As a layer of water is kept between all components, including the transducer and the steel ring and between the steel ring and steel observation plate on top of that ring, it is common for the water to end up in the hole. Because of these factors, the acoustic impedance at the anomaly is determined by the properties of PZT, air, steel, and water at room temperature since these are the media through which the acoustic wave passes. More on the experimental setup can be found in Section 2. The acoustic impedance of each media is calculated using the following formula [16]. The results of this calculation can be seen in Table 2.  
\[Z=density*velocity \tag{1}\]  
Using the values from Table 2 and the formula for the reflectance of a wave at the interface between two media as the wave travels in a nearly straight path, we can approximate the expected reflection to be as follows. On entrance into the anomaly leaving the transducer, we examine both the scenario where the wave travels from PZT into water and where the wave travels from PZT into air:  
\[PZT-Water=\frac{(Z_{w}-Z_{p})}{(Z_{w}+Z_{p})}=\frac{(31.5\times 10^{6}-1.481 \times 10^{6})}{(31.5\times 10^{6}+1.481\times 10^{6})}\approx-0.91 \tag{2}\]  
\[PZT-Air=\frac{(Z_{a}-Z_{p})^{2}}{(Z_{a}+Z_{p})^{2}}=\frac{(31.5\times 10^{6}-398. 89)}{(31.5\times 10^{6}+398.89)}\approx-1.00. \tag{3}\]  
Figure 3: Depiction of an acoustic wave traveling through anomaly in a solid material. The white lines of the figure depict the wave prior to interaction with the anomaly and after diffraction by anomaly. The black lines depict the signal captured by the sensor, represented as element waves. The horizontal dotted line labeled X denotes the axis along which the receiver sweeps/scans the specimen. The vertical dotted line labeled Z denotes the axis of propagation of the acoustic signal. We can interpret this calculation as meaning that at least 91% of the wave as it enters the anomaly will reflect back toward the emitter. For the incident portion of the wave which enters into the hole, we then consider the acoustic impedance of this wave as it exits the hole toward the steel observation plate:  
\[Steel-Water=\frac{(Z_{\mathrm{s}}-Z_{w})}{(Z_{\mathrm{s}}+Z_{w})}=\frac{(46.63 \times 10^{6}-1.481\times 10^{6})}{(46.63\times 10^{6}+1.481\times 10^{6})}\approx 0.938 \tag{4}\]  
\[Steel-Air=\frac{(Z_{\mathrm{s}}-Z_{\mathrm{a}})^{2}}{(Z_{\mathrm{s}}+Z_{\mathrm{ a}})^{2}}=\frac{(46.63\times 10^{6}-398.89)}{(46.63\times 10^{6}+398.89)} \approx 1.00. \tag{5}\]  
From the above calculations, we are able to infer that upwards of 93.8% of the wave which was transmitted through the hole will be reflected at the anomaly as the wave will either travel between steel and water or steel and air. This means that nearly all of the signal will reflect at the location of the hole, so nearly all of the wave will be bending around the obstacle at the center. This will lead to evidence of an acoustic shadow behind the hole. The nature of this acoustic shadow is determined by the size of the obstacle, the wavelength of the signal, and the distance to the observation point [17]. As mentioned in the introduction, data for this experiment are captured with two frequencies, 5 MHz and 1 MHz. The steel specimens being probed also contain 2 possible anomaly sizes, 6.47 mm in the larger anomaly cases and 2.17 mm in the smaller anomaly cases. These specimens are always observed at a distance of 6.47 mm, on the other side of the steel observation plate. Diagrams estimating the effect of these configurations on the observation of the acoustic shadow are depicted in Figure 4. These diagrams depict the anticipated acoustic shadow which will occur directly over the obstacle depicted in Figure 3 as the obstacle blocks a portion of the received signal.