## 3 Experimental Analysis
### Theoretical Background
#### 3.1.2 Room Modes  
The sound propagation in any acoustic enclosure follows different wave characteristic phenomena like reflection, scattering, diffraction, and interference. Such a complex interaction of numerous waves is characterized through the acoustical wave equation [5]. The frequencies corresponding to the eigenvalues of the acoustic wave equation can form standing waves inside the room to create a resonant behavior leading to non-uniform distribution of reflection power and extended reverberation [5; 43; 44]. These frequencies are often referred to as room modes or eigenfrequencies.  
According to [5; 43], at low frequency ranges, the number of resonant frequencies will be small, and they can be excited individually. Hence, the room response will be quite irregular and anisotropic for these frequencies. When we move towards the higher frequencies, the eigenvalues are densely spaced, so they cannot be independently excited. Even though the higher frequencies contribute to the reflected sound pressure, the lack of independent resonance combined with increased scattering makes them relatively uniform and less prominent compared to the lower frequencies. Hence, in a typical room response,we expect high reflection powers with some resonant peaks for low (<300 Hz) to mid (300 to 600 Hz) audible frequencies and decaying magnitude towards the high (>600 Hz) frequencies. The cross-over frequency [5; 43] that separates the resonant low-frequency response and the high-frequency diffused reflections is termed as Schroeder frequency (\(\
u_{5}\)). It can be calculated using the empirical formula  
\[\
u_{5}\approx 2000\sqrt{\frac{T_{60}}{\Delta}} \tag{16}\]  
where \(\Delta\) is the room volume. From the dimensions and \(T_{60}\) of the test rooms, (16) gives \(\
u_{5}\approx 184\) Hz and \(\approx\)169 Hz for Room-1 and Room-2, respectively.  
For a rectangular enclosure, we can calculate the eigenvalues of the wave equation [5; 42; 43; 44] as  
\[\
u_{n_{x}n_{y}n_{z}}=\frac{c}{2}\sqrt{\left(\frac{n_{x}}{l_{x}}\right)^{2}+ \left(\frac{n_{y}}{l_{y}}\right)^{2}+\left(\frac{n_{z}}{l_{z}}\right)^{2}} \tag{17}\]  
where \(\{n_{x},n_{y},n_{z}\}\) are non-negative integers and \(l_{x}\times l_{y}\times l_{z}\) are the room dimensions. When two of \(\{n_{x},n_{y},n_{z}\}\) equals zero, the solution of (17) gives the axial modes which are considered to be stronger with low decay rates compared to other modes [42]. We can calculate the tangential modes with two non-zero integers in \(\{n_{x},n_{y},n_{z}\}\) and oblique modes by substituting all non-zero integers in \(\{n_{x},n_{y},n_{z}\}\).  
Figure 2 shows the room mode distribution in Room-1 and Room-2. The axial and tangential modes are calculated from (17), and the line heights in Figure 2 represent the number of resonances occurring at a frequency since different \(\{n_{x},n_{y},n_{z}\}\) combinations can result in the same \(\
u_{n_{x}n_{y}n_{z}}\) frequency. The axial modes were given a higher nominal weight [44] while calculating this distribution due to their inherent prominence. Theoretically, an empty rectangular room of the same dimensions should replicate this trend in their frequency response. However, in a real room environment, the interference of normal modes of different decay rates [44] and the influence of inhomogeneous surfaces and source directivity alter the assumptions behind (17). Therefore, the real room response may vary from the predicted distribution.  
For practical validation of the real acoustic phenomenon, we will use the power response generated using the proposed technique to identify the variations in the room mode distribution and modal decays compared to the above theoretical expectations.  
Figure 2: Room mode distribution in (**a**) Room-1 (**b**) Room-2.