# An Experimental Study of the Performance of a Crossed Rib Diffuser in Room Acoustic Control
## 2 Specification and Acoustic Properties of CRD
### Absorption Characteristics  
This subsection evaluates diffuse field absorption coefficients for R-1, R-2 and the CRD. Measurements were conducted in a reverberation room of 212 m\({}^{3}\) (Technical Research  
Figure 4: Random-incidence scattering coefficients of various 1D periodic rib diffusers (R-1, R-2, R-1-W and R-2-W) and the CRD. Figure 3: 1/5 scale sample of CRD surrounded with a plastic border for scattering coefficient measurement. Institute, Hazama Ando Corp.) with a projected surface area of the test specimens of 10 m\({}^{2}\). As a sound source signal, the swept-sine signal of 30 s signal length was used. Other settings were in accordance with JJS A 1409 (ISO 354 [22] compatible). Here, to address the potential for transforming the CRD to a broadband absorber by inserting a sound-absorbing layer between the upper and lower ribs, we also manufactured a CRD with a sound-absorbing layer, CRD-A. Then its sound absorption performance was measured. The inserted sound-absorbing layer was a 0.4 mm thick, permeable, high-density polyester fabric [23]. Figure 5 shows the status of laying a sound-absorbing layer on R-1 (Upper) and of the sound absorption measurement of the CRD-A. As a reference, a statistical sound absorption coefficient of the sound-absorbing layer with a 21 mm air layer was calculated using a theoretical model for a single-leaf permeable membrane sound absorber [24]. As shown in [23], the theoretical model can accurately predict the absorption performance of a thin porous layer with a rigid-backed air cavity. As the ribs behind the sound-absorbing layer can play the role of a honeycomb structure [25], we calculated the statistical sound absorption coefficient of a single-leaf permeable membrane absorber with two assumptions for wave propagation in the air layer, i.e., the extended reaction (ER) and the local reaction (LR). The specific acoustic impedance ratio \(z\) of a single-leaf permeable membrane absorber is calculated using [24]  
\[z=\big{(}\frac{\rho_{0}c_{0}}{R}+\frac{\rho_{0}c_{0}}{i\omega M}\big{)}^{-1}- \frac{i}{\cos\theta}\cot(kL\cos\theta), \tag{1}\]  
where \(\rho_{0}\) and \(c_{0}\) denote air density and the speed of sound, respectively. Therein, \(R\) and \(M\) represent the flow resistance and the surface density of the sound-absorbing layer, respectively. In addition, \(L\) and \(\theta\) are the length of air layer and the sound-incident angle. \(\omega\) and \(k\), respectively, denote the angular frequency and the imaginary unit \(i^{2}=-1\). The statistical absorption coefficient with ER assumption was calculated by substituting Equation (1) into the following equation:  
\[\alpha_{statistical}=\frac{\int_{0}^{\pi/2}\alpha_{\theta}\sin\theta\cos \theta d\theta}{\int_{0}^{\pi/2}\sin\theta\cos\theta d\theta}, \tag{2}\]  
with  
\[\alpha_{\theta}=\frac{4\text{Re}[z]\cos\theta}{(\text{Re}[z]\cos\theta+1)^{2} +(\text{Im}[z]\cos\theta)^{2}}. \tag{3}\]  
For this study, we calculated the theoretical absorption coefficients with \(M\) = 0.20 kg/m\({}^{2}\) and \(R\) = 1050 Pa s/m [23]. Additionally, \(c_{0}\) and \(\rho_{0}\) were set to 343.7 m/s and 1.205 kg/m\({}^{3}\), respectively. The statistical absorption coefficient with LR assumption was assessed by substituting \(\theta\) = 0 into Equation (1). Figure 6 presents a comparison of diffuse field absorption coefficients of R-1, R-2, the CRD and the CRD-A; and the statistical absorption coefficients for a single-leaf permeable membrane absorber based on ER and LR assumptions. R-1 had almost no sound absorption performance, whereas R-2 showed absorption coefficient values of around 0.2 at frequencies higher than 2 kHz, with a gentle peak at 3.15 kHz. The CRD has absorption coefficients of around 0.2 at frequencies higher than 1 kHz, presenting better sound absorption performance of the CRD than R-1 and R-2. Furthermore, CRD shows a peak of the absorption coefficient at 1.6 kHz, which is lower frequency than that of R-2.