## 3 Experimental Activity and Results
### Analytical Acoustic Analysis of Combustion and Post-Combustion Chambers  
A simplified test was designed to investigate the acoustic properties of the HRE cavities, in which the combustion chamber cavity was connected to a modified volume representative of the post-combustion chamber (equivalent simple tube representing the cavity equipped with the nozzle). Based on the type of the acoustic boundary condition different cavity resonance frequencies and modes (Figure 10) are possible [20]. For instance, for a tube with an open end and a closed end longitudinal standing waves with frequencies equal to \((2n-1)c/(4L)\) can be established whereas for a tube closed at both ends or open at both ends, the wavelength associated with the fundamental frequency is twice the tube length, so, the natural frequencies will be \(nc/(2L)\). Specifically, \(n\) is the mode index, \(L\) is the effective acoustic length, and \(c\) is the speed of sound. As a result of the simple geometry, the effective acoustic length of the combustion chamber matches the geometric length, while the effective acoustic length of the post-combustion chamber can be set equal to the distance between the left side and the nozzle throat, less approximately one-half of the converging nozzle length [23]. The computational fluid dynamics (CFD) analysis carried out as part of the HYPROB project revealed a slight difference between the average temperature of the chamber and that of the post-combustion  
Figure 8: Sound pressure level 1000 N hybrid rocket engine (22 m and 35\({}^{\circ}\)). Figure 9: CIRA 1000 N HRE section [13]. chamber (chamber temperature is approximately 3 percent higher than post-combustion chamber). Therefore, the sound speed was assumed to be approximately constant and is estimated by averaging the values measured in the two chambers. The pipe eigenfrequencies for various boundary conditions were then evaluated and overlaid on the jet noise spectrum in Figure 11. The blue dotted line and the red dotted line indicate the chamber eigenfrequencies for the open-open (o-o) and closed-open (c-o) boundary conditions, respectively, while the blue and red stars represent the eigenfrequencies corresponding to the post-chamber modes with c-c and c-o boundary conditions, respectively. Due to the size of the chambers (the length of the combustion chamber is an integer multiple of the equivalent representative volume length of the post-combustion chamber), an acoustic coupling pattern occurs, and the high efficiency of sound transmission could be the consequence of such interaction. In order of frequency, the mode couplings involve:  
* The first post-combustion chamber longitudinal mode closed-open with the third combustion chamber longitudinal mode closed-open;
* The first post-combustion chamber longitudinal mode closed-closed with the fifth combustion chamber longitudinal mode open-open;
* The second post-combustion chamber longitudinal mode closed-open with the eighth combustion chamber longitudinal mode closed-open;
* The second post-combustion chamber longitudinal mode closed-closed with the tenth combustion chamber longitudinal mode open-open. Figure 11: SPL spectra compared to the combustion chamber and post-combustion chamber eigenfrequencies, for various boundary conditions. Figure 10: First acoustic eigenmode for various boundary conditions. The analytical solutions are collected and compared with the experimental values in Table 2. Included in this table are the percentage relative errors, computed as:  
\[\%Error=\frac{|f_{Exp}-f|}{f_{Exp}}100 \tag{7}\]  
In a conventional liquid propellant rocket engine, the acoustic coupling involves the interaction between the modes of the injector system and the combustion chamber, while in a hybrid rocket engine the same occurs between the combustion chamber and the post-combustion chamber.