# Effect of Subglottic Stenosis on Expiratory Sound Using Direct Noise Calculation
## 3 Results
### Neutral Glottis
#### 3.1.2 Sound Spectrum under Neutral Glottis
At an SGS blockage of 96%, a pure SGS hole tone of 816 Hz is excited at 100 mL/s. At 300 mL/s, the spectrum shows tonal peaks but is not pure. The major peak in the spectrum is the second harmonic of the closed-open XT. At 500 mL/s, a pure tone of 2357 Hz is excited, which is close to the 3 rd harmonic of the closed-open XT. The spatial waveform shows a wavelength that is slightly less than \(L_{\rm{vt}}\). The pressure node is downstream of the glottis exit, which indicates that the open-open vocal tract mode is not excited. At 700 mL/s, a strong pure tone is excited at 3286 Hz, which is very close to the overlapping frequencies of the 4 th harmonic of the closed-open XT and the 3 rd harmonic of the open-open VT. In this case, vortices are shed at the exit of the stenosis, and the wall pressure shows a 3/2 wavelength standing wave in the vocal tract (shown later in Figure 10a), corresponding to an open-open acoustic mode and a wavelength of 10 cm. The cavity length is roughly 2.5 cm, which is a 1/4 wavelength. Thus, a 7/4 wavelength standing wave is formed in the extended tract from the SGS to the mouth, with the glottal exit (x = 0 in Figure 1b) as a pressure node. This suggests that the acoustic modes of both the open-open VT and the closed-open XT are active in this case, which is possibly due to the specific vocal tract length and the SGS distance. At 1000 mL/s, a strong pure tone of 2455.5 Hz is excited, corresponding to the third harmonic of the XT with the closed-open end condition. The acoustic mode is a 5/4 wavelength standing wave, similar to the 500 mL/s case. The excited mode at 1000 mL/s is lower than that at 700 mL/s. This is probably due to the increase in the vortex convection speed with flowrate causing the coupling between the hole tone and the tract resonance to settle at a lower vortex mode. Inspection of the flow field confirms this as the number of vortices between the SGS and the glottis changes from two to one. The readers are referred to references [34; 35] for a more detailed and illustrated presentation of the nonlinear coupling phenomenon between the vortex mode of the jet-obstacle system and the acoustic mode of the resonator. Briefly, the vortex shedding frequency (\(f_{\rm{v}}\)) of the jet scales with the convection speed and vortex mode in the form \(f_{\rm{v}}\approx N_{\rm{v}}v_{\rm{C}}/L_{\rm{v}}\), where \(N_{\rm{v}}\) is the number of vortices between orifice and obstacle, \(v_{\rm{C}}\) is the vortex convection speed, and \(L_{\rm{v}}\) is the vortex travelling distance. Lock-in occurs when the shedding frequency is in the proximity of an acoustic mode of the resonator represented by \(f_{\rm{r}}\in\left\{f_{\rm{r}}^{1},f_{\rm{r}}^{2},f_{\rm{r}}^{3},\cdots,f_{\rm{ r}}^{n}\right\}\), where the superscript numeral represents the order of the acoustic modes. As the convection speed increases (roughly linearly) with the jet speed, the shedding frequency tends to increase and eventually deviates too much from the acoustic mode to stay locked-in. Before it reaches the next higher acoustic mode, it is possible for the vortex shedding to switch to a lower mode to lock in to a lower acoustic mode if a solution exists for the coupling \(f_{\rm v}\approx f_{\rm r}\) at an intermediate flow speed.