# Mechanism Analysis of the Influence of Seat Attributes on the Seat Dip Effect in Music Halls
## 3 Results and Discussion  
### Seat Height and Seat Spacing  
In previous classical studies [1,2], when \(h\approx\lambda/4\), where \(\lambda\) represents the sound wavelength at a specific frequency, the SPLs at the receivers well above the surface of the seat top were measured to be lowest and the corresponding specific frequency was just the dip frequency \(f_{0}\). This was caused by the classical interference between the grazing direct sound and the vertically reflected sound waves inside the seat gaps near the surface of the seat top. In the first numerical case, investigations were carried out with different seat heights with a fixed seat spacing of 0.8 m to check the SDE phenomenon. Figure 4a presents the corresponding results of the average \(RL\) at receivers R4-R6, where the SDE phenomenon was more distinct. When \(h\) is 1.1 m, 0.9 m, 0.7 m and 0.5 m, the theoretical \(f_{0}\) should be 80 Hz, 95 Hz, 120 Hz and 170 Hz, respectively, according to the quarter wavelength theory above. For comparisons, as shown in Figure 4a, the corresponding \(f_{0}\) in the simulations was about 80 Hz, 90 Hz, 110 Hz and 130 Hz, respectively, and was close to the theoretical values but had a noticeable shift to lower frequencies when the seat height \(h\) became smaller than the seat spacing \(w\).  
Figure 3: Illustration of the SDE: (**a**) relative level spectra; (**b**) color maps of the SPL spatial distributions at a dip frequency \(f_{0}\) of 105 Hz. Seats and floor are rigid.  
Another numerical case was conducted on the SDE phenomenon at a different seat spacing \(w\) with a fixed seat height of 0.8 m, results of which are presented in Figure 4b. It was shown that, when \(w\) was 0.5 m, 0.7 m and 0.8 m, the simulated \(f_{0}\) was almost the same with a theoretical value of 105 Hz for the seat height \(h\) of 0.8 m, but when \(w\) was 1.1 m or 1.3 m, and thus larger than the seat height, a noticeable shift to a lower frequency of 85 Hz or 80 Hz was observed, respectively, as well as those in the first numerical case above. This phenomenon indicates that, when the seat height \(h\) becomes smaller than the seat spacing \(w\), the dip frequency \(f_{0}\) may be shifted to lower frequencies compared to values from the quarter wavelength theory [1; 2]. This may be caused by the interference between the horizontal reflections of the oblique direct sound on the seat backs. As the seat height becomes smaller or the seat spacing larger, the oblique direct sound from the source that can travel into the seat gaps becomes stronger, and leads to stronger horizontal reflections on the seat backs, and thus a stronger interference between those reflections, whose interference frequency is controlled by the seat spacing \(w\). When \(h\) is smaller than \(w\), the interference between the horizontal reflections becomes stronger than the interference between the grazing direct sound and vertical reflections, where the frequency of the former interference is clearly lower than that of the latter interference. This leads to the observed lower frequency shift of dip frequency \(f_{0}\) in this situation, and can also explain the \(RL_{0}\) decreasing with \(w\) increasing, as shown in Figure 4b.  
Figure 4: Spectra of average \(RL\) at receivers R4-R6 for investigation of influence on the SDE from (**a**) seat heights (\(w=0.8\) m, seats and floor are rigid); (**b**) seat spacing (\(h=0.8\) m, seats and floor are rigid); (**c**) seat back absorption (\(h=1.1\) m and \(w=0.8\) m) and (**d**) seat cushion absorption (\(h=1.1\) m and \(w=0.8\) m).