## 3 Experimental Activity and Results
### Eldred Model Results  
In order to compare the experimental data with the theoretical predictions, the narrow-band sound pressure level was converted into one-third octave bands, as shown in Figure 12. In addition, A-weighting was applied to the SPL in an effort to describe the effective level perceived by the human ear, which is less sensitive to low frequencies. The effect of large reflective surfaces, such as the concrete ground parallel to the tested engine, should be included in the prediction by estimating the additional noise to the considered point. Therefore, in this work the sound pressure level was adjusted based on the following considerations. The sum of two identical signals (e.g., the direct and the reflected waves) with the same phase (coherent), produces an SPL of +6 dB compared to the single signal's SPL. However, in this work the relative phase between the direct \begin{table}
\begin{tabular}{c c c c} \hline \hline
**\# Coupling** & **Experimental** & **Analytical** & \\
**Frequency** & **Frequencies (Hz)** & **Frequencies (Hz)** & **Error (\%)** \\ \hline
1 st & 3364 & 3360 & 0.12 \\
2 nd & 6725 & 6721 & 0.06 \\
3 rd & 10,127 & 10,081 & 0.45 \\
4 th & 13,528 & 13,442 & 0.64 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Natural frequencies of HRE chambers, with relative error. signal and its reflection is not known, so in general, there could be both a sum of in-phase signals (+6 dB SPL), and the opposite effect, that is, a total cancellation when the phases have a difference of 180 degrees. Thus, considering fully incoherent sources represents the only way to describe the reflection phenomenon. More specifically, the summation of two incoherent signals will result in an SPL of +3 dB with respect to the single signal's SPL. To conclude, it is reasonable to expect an increase in sound field levels of 3-6 dB [11], assuming perfect reflection (no absorption by the ground). Figure 13 shows the comparison between the predicted noise results and the experimental data. For the reasons explained above, the expected experimental SPL range (without reflection component) is obtained by subtracting 3 and 6 dB from the original experimental curve (gray band). The solid and dashed black lines represent the predicted curve (obtained by [PERSON]'s procedure) assuming 1% and 0.5% acoustic efficiency values. [PERSON]'s empirical method is estimated to have an accuracy of \(\pm\)4 dB in terms of overall sound pressure level. Analysis of the case study curves confirms that these values are within these limits. In particular, considering the case of completely incoherent sources (OASPL-3 dB) in relation to the prediction for the HRE assuming an acoustic efficiency of 1% (upper limit), it can be stated that the trend of the sound pressure spectrum curves is quite similar, with a \(\Delta\)OASPL of 2.2 dB for the un-weighted SPL and a \(\Delta\)OASPL of 2.68 dB for the A-weighted SPL case. For more details, see Table 3:  
Figure 12: The 1/3 octave Un-/A-weighted SPL 1000 N hybrid rocket engine (22 m and 35°). Figure 13: Expected experimental SPL range (gray shape) compared to predicted SPL with 1% (solid black line) and 0.5% (dashed black line) acoustic efficiency. (**a**) Un-weighted SPL. (**b**) A-weighted SPL. Although there is a large difference between the expected experimental SPL and [PERSON]'s predicted SPL at low frequencies (Figure 13), the predicted OASPL is essentially correct. The reason is that the energy at these frequencies is very low (the graph is in dB scale) and an error would not affect the total energy content. From a physical point of view, such a mismatch could be explained by the occurrence of wave diffraction phenomena, which normally result in constructive and destructive interference. They occur when waves pass through a structure whose dimensions are close to the wavelength of sound. In the test case, at 160 Hz, the corresponding wavelength of 2 m becomes comparable to the size of the box.