## 3 Theory
### Estimation of the Inherent Parameters \(T_{ng}\), \(T_{g}\) and \(C\)
#### 3.2.1 The Angle-Dependent Absorption Coefficient  
For each sector, representative absorption coefficients (\(\alpha_{ng}\), \(\alpha_{g}\)) and a representative number of modes (\(\Delta N_{ng}\), \(\Delta N_{g}\)) have to be determined. It is assumed that the surface impedance of the ceiling absorber is known or can be estimated. Several types of commercial software' are available today for calculating the angle-dependent surface impedances [57; 58] for different types of absorbers. In this study, only suspended ceilings of porous material were investigated. For porous absorbers, the surface impedance \(Z(f,\theta)\) can be calculated by applying empirical models if the air flow resistivity is known. In this case [PERSON]'s model was used [59]. An extended reaction is assumed when calculating \(\alpha_{ng}\) and \(\alpha_{g}\). The angle-dependent absorption coefficient for a plane sound wave impinging on a plane infinite surface is given by  
\[\alpha(f,\theta)=1-\left|\frac{Z(f,\theta)\cos(\theta)-\rho_{0}c_{0}}{Z(f, \theta)\cos(\theta)+\rho_{0}c_{0}}\right|^{2} \tag{21}\]  
where \(Z(f,\theta)\) is the surface impedance at incidence angle \(\theta\), \(\rho_{0}\) is the density of air, and \(c_{0}\) is the speed of sound. The surface impedance for an extended reaction is calculated as [60]  
\[Z(f,\theta)=\frac{Z_{c}k}{k_{x}}\left[\frac{-jZ_{0}\cot(k_{x}d)+Z_{c}\frac{k}{ k_{x}}}{Z_{0}-jZ_{c}\frac{k}{k_{x}}cot(k_{x}d)}\right] \tag{22}\]  
where \(k\) is the wave number in the absorber, \(k_{x}=\sqrt{k^{2}-k_{0}^{2}\sin^{2}(\theta)}\) is the normal component of \(k\), \(k_{0}\) is the wave number in air, \(d\) is the thickness of the absorber, and \(Z_{c}\) is the characteristic impedance of the absorber. The backing impedance \(Z_{0}\) is given by  
\[Z_{0}(f,\theta)=-j\bigg{(}\rho_{0}c_{0}\frac{k_{0}}{k_{x}}\bigg{)}\cot(k_{0}d _{0}\cos\theta) \tag{23}\]  
where \(d_{0}\) is the depth of the air cavity behind the absorber.  
Figure 2: Illustration of the grazing and non-grazing sectors.  
The characteristic impedance for the absorber \(Z_{c}\) is calculated by [PERSON]'s model according to  
\[Z_{c}=\rho_{0}c_{0}\left[1+0.070\left(\frac{f}{\sigma}\right)^{-0.632}-j0.107 \left(\frac{f}{\sigma}\right)^{-0.632}\right] \tag{24}\]  
and wave number  
\[k=\frac{\omega}{c}\left[1+0.109\left(\frac{f}{\sigma}\right)^{-0.618}-j0.160 \left(\frac{f}{\sigma}\right)^{-0.618}\right] \tag{25}\]  
The only material parameter needed for [PERSON]'s formula is the air flow resistivity \(\sigma\) of the porous material. [PERSON]'s formula is an improvement of the Delany and Bazley model [61]. Another modification of the Delany and Bazley model has been developed by [PERSON] [62].  
The extended reaction (the angle-dependent impedances) is of particular importance for accurate estimation at low frequencies. This is illustrated in Figure 3 where local and extended reactions are compared for the reverberation time \(T_{20}\) measured in a sparsely furnished room with dimensions 7.56 m \(\times\) 7.30 m \(\times\) 3.50 m and with a 15 mm thick absorbent ceiling at a mounting height of 200 mm (case 4 in Section 4). The figure shows the results using extended vs. local reaction in the model. Considerable deviation at low frequencies (125 Hz and 250 Hz) appears.