# Tunable Perforated Panel Sound Absorbers for Variable Acoustics Room Design
## 3 Methods
### Ray Tracing Method  
To study the influence of using the above solutions as part of a variable acoustic system, simulations for a case study room were performed by using a self-developed MATLAB v. 2021b code that implements the ray tracing method. While several numerical approaches have been recently proposed for the acoustic simulation of rooms with perforated panel absorbers, either using the finite element method (FEM) [21; 22] or the finite-difference time-domain (FDTD) method [23; 24], the ray tracing approach was herein chosen for the sake of simplicity and computational efficiency, as is usual in room acoustics design. The implementation carried out in this work followed a similar procedure to that used in a previous work by the authors [25], which proved to be successful in the simulation of a conceptual auditorium.  
In the present work, a room with dimensions of 5 m \(\times\) 9 m \(\times\) 3 m, with a total volume of 135 m\({}^{3}\), was prepared using SketchUp 3D design software, and the simulations were performed using around 50,000 rays to obtain impulse responses of 2 s length at different receiver positions when an omnidirectional source is placed in a representative point. From the impulse response data, it was straightforward to calculate classical room-acoustic parameters, such as the Reverberation Time (RT), the Definition (D50), and the Clarity (C80), whose definitions can be found in several classic reference books on architectural acoustics, such as the one by [PERSON] [26]. Simulations were performed for different occupancy levels both without any absorber (except for reference lining materials on each surface) and with the TS-A solution described in Section 2, this being positioned in the rear wall of the room and in one-third of the surface of the lateral adjacent walls (the standard implementation solution of absorbing material to control reverberation in a room used for lectures or classes, for example). Insomuch as the model reproduces a diffuse field scenario, it was necessary to calculate the diffuse field sound absorption coefficient of the proposed solutions from the measured normal incidence sound absorption data by using well-known formulas represented by equations (11.89) to (11.91), which can be found in Chapter 11 in [3]. In this regard, the particular solution (TS-A) chosen to highlight the tuning capabilities of the proposed systems in diffuse field conditions was approximated as locally reacting as the dimensions of the perforations are much smaller than the smaller wavelength of interest (i.e., the sound propagation in each pore depends only on the pressure of the air above the pore). Corresponding values of these sound absorption coefficients will be given in Section 4.2. For the remaining lining materials, absorption data from the literature were used [2], as summarized in Table 2. Figure 2 shows the resulting geometry along with the identification of all the lining materials and the location of the source and receivers.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Standard Solution**} & \multirow{2}{*}{**125 Hz**} & \multirow{2}{*}{**250 Hz**} & \multirow{2}{*}{**500 Hz**} & \multicolumn{3}{c}{**Frequency**} & \multirow{2}{*}{**4 kHz**} & \multirow{2}{*}{**8 kHz**} \\  & & & & & & & \\ \hline Occupied seat & 0.18 & 0.24 & 0.28 & 0.33 & 0.37 & 0.39 & 0.42 \\ (m\({}^{2}\) per unit) & 0.02 & 0.02 & 0.03 & 0.04 & 0.06 & 0.08 & 0.08 \\ Floor in wood parquet on & & & & & & & \\ concrete (\(\alpha\)) & 0.04 & 0.04 & 0.07 & 0.06 & 0.06 & 0.07 & 0.07 \\ Plastered masonry wall (\(\alpha\)) & 0.01 & 0.02 & 0.02 & 0.03 & 0.04 & 0.05 & 0.05 \\ Suspended ceiling (\(\alpha\)) & 0.29 & 0.25 & 0.25 & 0.23 & 0.19 & 0.19 & 0.18 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Sound absorption coefficients (\(\alpha\), \([-]\)) and sound absorption area of furniture elements (A, [m\({}^{2}\) per unit]) of the standard solutions (retrieved from [2]).