## 3 Theory  
### The SEA Model  
The sound field in a room with absorbent ceiling treatment is modelled as an SEA system consisting of two subsystems. One subsystem comprises non-grazing waves and the other comprises grazing waves. The term grazing refers to the angle of incidence towards the ceiling absorber. Thus, grazing comprises waves travelling almost parallel to the absorbent ceiling. The coupling loss factor between the two subsystems is related to the energy transfer from the grazing subsystem to the non-grazing subsystem. This energy transfer is most often due to the interior fittings in the room such as furniture, but could also be due to a tilting wall, for example. The back-transfer from the non-grazing to the grazing subsystem is neglected. The SEA model is illustrated in Figure 1.  
The power flow into the grazing (\(g\)) and non-grazing (\(ng\)) subsystem (\(\Pi_{ng},\Pi_{g}\)), as well as the dissipated power (\(\Pi_{ng,d},\Pi_{g,d}\)), are shown in Figure 1. The total energy in the subsystems are denoted as \(E_{ng}\) and \(E_{g}\), respectively. The power lost by the grazing subsystem to the non-grazing is represented by \(\Pi_{g,ng}\). Generally, a weak coupling is assumed, i.e., that the losses related to the coupling between the two system is less than the internal losses in the grazing and non-grazing subsystems [54]. Figure 1: The SEA model. In a room with non-uniform distribution of absorption, such as the rectangular room with a highly absorbent ceiling and the other surfaces almost reflecting, the energy decay is estimated by  
\[E(t)=E_{ng}(0)e^{-\omega\eta_{ng}t}+E_{g}(0)e^{-\omega\eta_{g}t} \tag{3}\]  
\(E_{ng}(0)\) and \(E_{g}(0)\) are the initial energies for the non-grazing and grazing subsystems, respectively. The loss factor in the non-grazing and the grazing subsystems are denoted as \(\eta_{ng}\) and \(\eta_{g}\), respectively. Using \(\Pi=\omega\eta E\) and, assuming that the coupling loss factor is negligibly small compared to the internal losses in the two subsystems, the energy ratio is given by  
\[E(t)=E_{ng}(0)(e^{-\omega\eta_{ng}t}+\frac{E_{g}(0)}{E_{ng}(0)}e^{-\omega\eta_{ g}t})=E_{ng}(0)(e^{-\omega\eta_{ng}t}+\frac{\eta_{ng}\Pi_{g}}{\eta_{g}\Pi_{ng}}e^{- \omega\eta_{g}t}) \tag{4}\]  
The condition in Equation (4) above is valid for a rectangular room with absorbent ceiling, but without furniture. Including furniture will lead to the introduction of a coupling loss factor related to the energy transfer from the grazing to the non-grazing sound field, see Figure 1. Replacing \(\eta_{g}\) in Equation (4) by \(\eta_{g}+\eta_{g,ng}\) where \(\eta_{g,ng}\) is the coupling loss factor, the absorbing and scattering effect of furniture can be accounted for. The coupling loss factor is further discussed in Section 3.2.5. As shown in [55], the ratio \(\Pi_{g}/\Pi_{ng}\) is approximately given by \(N_{g}/N_{ng}\), where \(N_{g}\) and \(N_{ng}\) are the number of modes in the grazing and the non-grazing subsystems, respectively. In geometrical acoustics, sound waves are often represented as rays with a certain sound intensity. Further, in room acoustical calculations, the reverberation time is a well-established parameter and normally the frequency depending on reverberation times are studied in frequency bands, usually octave bands. By converting Equation (4) into sound intensity, assuming octave band values and using the relation \(\Delta\Pi_{g}/\Delta\Pi_{ng}\approx\Delta N_{g}/\Delta N_{ng}\), and further introducing the reverberation time \(T\) using the relation \(\omega\eta=6\ln(10)/T\), we get  
\[I(t)=I_{ng}(0)(e^{-13.8t/T_{ng}}+\frac{T_{g}\Delta N_{g}}{T_{ng}\Delta N_{ng}} e^{-13.8t/T_{g}}) \tag{5}\]  
The procedure presented for a linear decay by [PERSON] and [PERSON] [27] is applied for the double sloped decay, as given by Equation (5). The steady-state condition at \(t=0\) gives the power balance  
\[W=\omega\eta\frac{I}{c}V \tag{6}\]  
where \(W\) is the input power and V is the room volume.