## 3 Theory
### Estimation of the Inherent Parameters \(T_{ng}\), \(T_{g}\) and \(C\)
#### 3.2.3 Estimation of \(\alpha_{ng}\) and \(\Delta N_{ng}\)  
To estimate \(\alpha_{ng}\), an intermediate step was used. This step includes the introduction of a weighted normalised absorption coefficient given by  
\[\alpha_{n}(f,\theta)=\frac{\alpha(f,\theta)\Delta N(f,\theta)}{\max(\alpha(f, \theta)\Delta N(f,\theta))} \tag{29}\]  
In this expression, \(\Delta N(f,\theta)\) is the number of modes as a function of frequency and angle, as given by Equation (28) replacing \(\theta_{g}\) with \(\theta\). The absorption coefficient \(\alpha(f,\theta)\) is the angle-dependent absorption coefficient given by Equation (21), assuming an extended  
Figure 4: Grazing sector in the wavenumber space. reaction. The non-grazing absorption coefficient \(\alpha_{ng}\) is given by Equation (21) for an angle (\(\theta_{ng}\)) corresponding to the maximum value in the distribution given by Equation (29). Examples of this distribution are given for case 1 in Section 4 and for the frequencies 250 Hz and 4000 Hz, see Figure 5. In the classical diffuse field assumption, the angle-dependent absorption coefficient is weighted by the factor \(\sin(2\theta)\), according to the Paris formula [63]. For comparison, the diffuse field weighting \(\sin(2\theta)\) is also shown. As can be seen in the figure, there is a bias between the classical approach and the distribution, according to Equation (29). The representative angle for the non-grazing absorption coefficient is somewhat higher compared to the \(\sin(2\theta)\). For the higher frequency, we see that the classical weighting corresponds to almost 45 degrees, as expected. The irregular shape at 250 Hz is due to the assumption of an extended reaction. Two approaches for determination of \(\Delta N_{ng}\) were used: a theoretical one and an empirical one. For the empirical approach, the number of non-grazing modes was determined by adjustment towards experimental results for several configurations where room dimensions and acoustical treatment and furnishing were varied. An approach using minimization of a cost function to perform a curve fitting is presented in [64]. In the empirical method, the upper and lower angles defining the non-grazing sector, see Figure 2, are given by \(\theta_{ng,\ lower}=\theta_{ng}(1-\Delta\theta)\) and \(\theta_{ng,\ higher}=\theta_{ng}(1+\Delta\theta)\), where \(\Delta\theta\) was estimated by comparison with measurements. Note that the angle of incidence \(=\pi/2-\theta\). Further, \(\theta_{higher}\) is restricted to be less than \(\pi/2\). The values for \(\Delta\theta\) is given in Table 1 for the octave bands 125 Hz to 4000 Hz. The number of modes in the non-grazing sector \(\Delta N_{ng}\) in Figure 2 is given by the repeated use of Equation (28) and is given by  
\[\Delta N_{ng}=\Delta N_{g}\big{(}\theta_{ng,upper}\big{)}-\Delta N_{g}\big{(} \theta_{ng,lower}\big{)} \tag{30}\] \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline Frequency Hz & 125 & 250 & 500 & 1000 & 2000 & 4000 \\ \hline \(\Delta\theta\) & 0.63 & 0.31 & 0.14 & 0.17 & 0.07 & 0.08 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Empirical determined limit parameter \(\Delta\theta\) for defining the non-grazing region. Figure 5: Distribution curves for the normalised weighted absorption coefficient according to Equation (29) for case 1 in Section 4.1. (Red) diffuse model, (blue) non-diffuse model. The theoretical approach involves calculating the number of modes for the non-grazing sector \(\Delta N_{ng}\) as  
\[\Delta N_{ng}=\frac{1}{\alpha_{ng}}\int\limits_{0}^{\pi/2}\alpha(f,\theta)N(f, \theta)d\theta \tag{31}\]  
where \(\alpha_{ng}\) is the absorption coefficient corresponding to the angle defined by the maximum in the weighted normalized absorption coefficient given by Equation (29). By knowing this non-grazing angle, see Figure 5 right, the non-grazing absorption coefficient \(\alpha_{ng}\) can be given by Equation (21).