# Deep Machine Learning for Path Length Characterization Using Acoustic Diffraction
## 1 Introduction
### Theory
As we see in the figures, theoretically, the acoustic shadow should be apparent in both of the 1 MHz cases as the point of observation falls well within the acoustic shadow depicted by the gray triangle, but may be more difficult to observe or even missed in the 5 MHz cases where the observation point falls nearly at the tip of the anticipated shadow. This is hinted at by the observation distance and width of the shadow shown by the dotted line in Figure 2:0  
In addition to the acoustic shadow, the divergence of the wave as it passes around the obstacle varies depending on the largest dimension of the obstacle and the observation point. When the distance between the observation point and the obstacle is much greater than the largest dimension of the obstacle, the diffraction is Fraunhofer diffraction [18; 19]. The further into the Fraunhofer range the observation point moves, the more Fraunhofer-like the observed wavefront becomes. When the distance to the observation point is comparable or less than the largest dimension of the obstacle, the diffraction is Fresnel diffraction [20]. As with the Fraunhofer range, the further into the Fresnel range the observation point moves, the more Fresnel-like the observed wavefront becomes. Similarly, as the observation point approaches the boundary between Fresnel and Fraunhofer diffraction, the wavefront is observed to be less Fresnel-like or less Fraunhofer-like. The problem with approximations in the Fresnel domain is that this form of diffraction is not a simple function. In the Fresnel range the shape and intensity of the diffraction pattern take many forms as the wave propagates downstream due to the increased scattering, while in the Fraunhofer \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Medium** & **Density (kg/m\({}^{3}\))** & **Velocity (m/s)** & **Impedance (Rayls)** \\ \hline Water & 1000 & 1481 & \(1.481\times 10^{6}\) \\ \hline Air & 1.204 & 331 & 398.89 \\ \hline Steel & 7850 & 5940 & \(46.63\times 10^{6}\) \\ \hline PZT & 7500 & 4200 & \(31.5\times 10^{6}\) \\ \hline \end{tabular}
\end{table} 
Table 2: Values for derivation and acoustic impedance of media involved in this experiment. range the shape and intensity remain rather constant [20]. This is shown in Figure 5 as the Fresnel regime, the region prior to the Fraunhofer diffraction limit (R), begins with a rectangular pattern and evolves into a pattern with many peaks as it approaches R. The calculations of the Fraunhofer limit (R) are displayed in Table 3, which are calculated using the following equation. \[R=D^{2}*\lambda, \tag{6}\]  
where D is the largest dimension of the anomaly  
The Fresnel diffraction pattern takes on many forms between the zones shown in this figure, only two Fresnel zones are shown here to reduce the complexity of this image [21]. On the other hand, the Fraunhofer regime maintains a very similar shape as it begins at R and moves further away. In this research, we observe anomalies in both the Fresnel and Fraunhofer regime, as can be seen by the change in proximity of the red dotted line in Figure 5. This figure depicts the changes in the diffraction pattern of an acoustic signal as the wave propagates away from the origin. Figure 4: This figure depicts the observation point (depicted as a dotted line) in relation to the anticipated acoustic shadow the gray triangle for the four configurations used in this experiment. This figure is a theoretical representation of the expected acoustic shadow as the wave begins to bend around an obstacle, represented by the black box, as more wavefronts, represented by the vertical partitions, pass the obstacle. Here we note the width of the acoustic shadow at the point of observation, which is decreasing from the left-most figure to the right-most figure.