## 2 General Description of the Model  
A general discussion of the model is presented in this chapter. The model is based on a statistical energy analysis (SEA) approach [53; 54]. The model addresses rectangular rooms with absorbent ceilings, i.e., rooms where the main contribution to the total absorption is related to the ceiling. A more precise requirement for this condition is given further on. Important considerations are, firstly, that the surface impedance of the absorbent ceiling, including the air cavity behind the absorber, has to be known, and secondly, that the absorbing and scattering effects of furniture and other interior fittings have to be estimated. A method for measuring the scattering effect is proposed in Section 3.2.5. This method takes into account the directional scattering of objects due to the orientation towards the ceiling. With the exception of the ceiling, other surfaces in the room are characterized by the statistical absorption coefficient. Further, added wall panels are defined by their statistical absorption coefficient, as measured according to ISO 354 [45]. The room acoustic parameters calculated are reverberation time \(T_{20}\) according to ISO 3382-2, speech clarity \(C_{50}\) in dB, and sound strength \(G\) in dB according to ISO 3382-1. Speech clarity is defined as  
\[C_{50}=10\log\left(\frac{\int_{0}^{0.05}p^{2}(t)dt}{\int_{0}^{\infty}p^{2}(t) dt}\right) \tag{1}\]  
where \(p(t)\) is the impulse response at the measurement point. Sound strength is defined as  
\[G=10\log\left(\frac{\int_{0}^{\infty}p^{2}(t)dt}{\int_{0}^{\infty}p^{2}_{10}(t) dt}\right) \tag{2}\]  
where \(p(t)\) is the impulse response at the measurement point and \(p_{10}(t)\) is the impulse response measured at 10 m in a free field. An omni-directional sound source is required for measurement of the acoustical parameters. The model comprises the following steps:  
Basic formulas are derived in Section 3.1 comprising  
Establish a general expression for the energy sound decay in a two-system SEA model. Express the total sound energy decay in the parameter sound strength \(G\) as defined in ISO 3382-1. From the expression for the total sound energy decay, derive an expression for the speech clarity \(C_{50}\) and the reverberation time \(T_{20}\). Estimation of the inherent parameters in the basic formulas are presented in Section 3.2 comprising  
Subdivide the total sound field into a grazing and non-grazing part where grazing refers to sound waves propagating almost parallel to the absorbent ceiling. Calculate the angle-dependent absorption coefficient, Section 3.2.1. Estimate the number of modes in the grazing subsystem as well as a representative absorption coefficient, Section 3.2.2. Estimate the number of modes in the non-grazing subsystem as well as a representative absorption coefficient, Section 3.2.3. Two approaches for estimation of the number of non-grazing waves were used: one empirical and one theoretical. Based on a 2-dim and 3-dim reverberation formula, estimate the reverberation times \(T_{g}\) and \(T_{ng}\) corresponding to the grazing and non-grazing subsystem, respectively. See Section 3.2.4. * By knowing \(T_{g}\) and \(T_{ng}\) and the number of modes in each subsystem, the energy ratio C for the grazing and non-grazing sound fields in the formula for sound strength \(G\) can be calculated. As an effect of the subdivision of the total sound field into a grazing and non-grazing part, the scattered and absorbed sound, due to objects such as furniture in the room, can be interpreted as a coupling loss factor between the two subsystems, see Figure 1. The coupling loss factor is reformulated as an equivalent scattering absorption area, denoted as \(A_{sc}\). A corresponding measurement method of \(A_{sc}\) is suggested. See further Section 3.2.5. As the distance \(r\) is included in the model, the room acoustic parameters as a function of distance can be calculated. However, in the calculations performed, a representative value of \(r\) is used. See Equation (41).