# Mechanism Analysis of the Influence of Seat Attributes on the Seat Dip Effect in Music Halls
## 2 Simulation  
From previous studies [7; 15], the reflections from walls and ceilings in a room can mask the SDE phenomenon, and the SDE is substantially related to the sound reflections from the seats and the floor. To simplify the problem and highlight the SDE phenomenon, a two-dimensional geometry with simplified acoustic boundary conditions, shown in Figure 1, was established to represent the section plane of classical music halls in the present study. The simulations were carried out with the acoustic FEM in the frequency domain and were implemented with the commercial software, COMSOL Multiphysics(r) [19]. In the numerical model geometry in Figure 1a, the room is defined as 24.5 m long and 4.5 m high with four boundaries (ground, ceiling, front wall, and back wall), containing 16 rows of simplified seats with the height \(h\) and the spacing \(w\) (representing the seat pitch in actual halls). A point source S is 4.5 m away from the front wall, at a height of 2.5 m above the ground, and radiates with a strength of 1 W/m\({}^{2}\). The receiving points R1-R6 are in the middle of the seat gap, at a height of 1.1 m above the ground. To simplify the problem, we assumed the ground to be perfectly reflecting with infinite impedance, and assumed the front wall, back wall and ceiling to be perfectly absorbing without sound reflections from these boundaries (which were implemented as Perfectly Matched Layers (PML) in numerical models) ensuring that the reflections were only from the ground and the seats to emphasize the SDE. The seats were simplified to vertical slats in the model. This configuration was used in previous scale models [2; 4] and calculation models [7], while the seat sound absorption could not be considered in the former plywood modelsand the sound field distribution and acoustic behavior between the seat gaps was not observed in the latter calculation model. In the numerical models in this paper, the seat back or the floor can be defined as rigid or with a different acoustic impedance. For the investigations of the seat height and seat spacing, the seats were set as rigid boundaries to separate the influence from the seat sound absorption. For investigations on the seat-back sound absorption, the seats were set with different acoustic impedances. It should be noted that the influence from the seat cushion sound absorption on the SDE was investigated by applying different impedances on the ground between the seats in the simplified model. In the present study, the surface sound absorption was illustrated by the normal incidence sound absorption coefficient \(\alpha\) (seat back absorption by \(\alpha_{\text{b}}\) and the seat cushion absorption by \(\alpha_{\text{c}}\)). A quantity of Relative Level (\(RL\)) was applied for the investigations on the characteristics of the SDE, which are defined as:  
\[RL=SPL_{2}-SPL_{1} \tag{1}\]  
where \(SPL_{2}\) and \(SPL_{1}\) are the Sound Pressure Levels (\(SPL_{\text{s}}\)) at each receiver when there are seats and no seats, respectively. At first, the numerical model was compared with the measurements of a \(1/10\) scale-model from [PERSON] and West [2] for validation. In the validation case, the geometry dimensions in the simulation were set to equal values of those in the scale-model (as shown in Figure 9 in Ref. [2]) and the sound absorption arrangement of the seats and floor in the simulation was set according to the scale-model conditions [2; 8], where the stage and the ground were set as rigid surfaces, and the seats were set as absorptive surfaces with a specific acoustic admittance of \(0.01\). The \(RL\) results of the simulation and measurements at a receiver of \(1.6\) m from the sound source in this validation case were compared in Figure 2. A good agreement can be observed between the simulation and measurements, except that in the measurements the \(RL\) dip was not identified as clearly as it was in the simulation due to the limited measurement data. This shows the validity of numerical models for the further SDE investigations in this paper.