## 3 Theory
### Estimation of the Inherent Parameters \(T_{ng}\), \(T_{g}\) and \(C\)
#### 3.2.5 Estimation of \(A_{sc}\)  
The sound-scattering effects of furniture and other objects in rooms will greatly influence the room acoustic parameters in rooms where the absorbent material is concentrated to the ceiling. Reverberation time \(T_{20}\) and speech clarity \(C_{50}\) will be particularly affected. Sound strength \(G\) will normally be less affected as it is related to the steady-state conditions and thus will not be sensitive to the distribution of the absorbent material. To quantify the scattering effect, the following procedure was used.  
In the terminology of SEA, the transfer of energy from the grazing to the non-grazing sound field is expressed in a coupling loss factor \(\eta_{g,ng}\). The power flow \(\Pi_{g,ng}\) from the grazing to the non-grazing subsystem is given by  
\[\Pi_{g,ng}=\omega\eta_{g,ng}E_{g} \tag{35}\]  
where \(\eta_{g,ng}\) is the coupling loss factor from the grazing to the non-grazing subsystem and \(E_{g}\) is the energy in the grazing subsystem.  
The coupling loss factor \(\eta_{g,ng}\) can be estimated in a rectangular room with a highly absorptive ceiling. It is assumed that the two-system SEA model is valid for the sound field in the room, both with and without scattering objects (furniture) present. This is very often the case in rooms with absorbent ceiling treatment, as it is really difficult to create isotropic conditions in these types of rooms.  
The coupling loss factor is then given by  
\[\eta_{g,ng}=\eta_{g,with\ obj}-\eta_{g,without\ obj} \tag{36}\]  
where \(\eta_{g,with\ obj}\) is the grazing loss factor with objects in the room and \(\eta_{g,\ without\ obj}\) is the grazing loss factor without objects in the room. These loss factors are determined from the reverberation time \(T_{20}\), i.e., the late part of the decay curves in the room with and without objects. The relation between the reverberation time \(T\) and the loss factor is given by  
\[\eta=\frac{6 ln10}{\omega T} \tag{37}\]  
In a two-dimensional sound field, an equivalent scattering absorption area can be defined as [55]  
\[A_{sc}=\frac{\pi\omega V}{c}\eta_{g,ng} \tag{38}\]  
where \(c\) is the speed of sound and \(V\) is the room volume.  
Combining Equations (36)-(38) gives the equivalent scattering absorption area for objects as  
\[A_{sc}=0.127V\bigg{(}\frac{1}{T_{20,with}}-\frac{1}{T_{20,without}}\bigg{)} \tag{39}\]  
where \(T_{20,\ with}\) and \(T_{20,without}\) are the reverberation times in the room with ceiling absorber, with and without objects, respectively. Equation (39) assumes that the late reverberation time \(T_{20}\) in a room with a highly absorptive ceiling is determined by a two-dimensional sound field. The measure \(A_{sc}\) is affected by the sound scattered into the ceiling and by the absorption of the objects. This measure is similar to the equivalent absorption area used in Sabine formula. It is used in the same way in Equation (34).  
Of course, the \(A_{sc}\) will depend on the ceiling absorption properties. However, if the mean absorption coefficient of the ceiling absorber, for the mid and high frequencies, is larger than about 0.7, we will obtain a reasonable estimation of \(A_{sc}\) that can be used in most common situations of rooms with absorbent ceilings [65].  
The \(A_{sc}\) for the investigated furniture configurations were measured according Equation (39) and are further discussed in the Section 5.