# Effect of Subglottic Stenosis on Expiratory Sound Using Direct Noise Calculation
## 1 Introduction
### Mechanisms of Flow-Sustained tones  
Flow-sustained tones involve unstable flows and some form of an amplification-feedback loop. [PERSON] [24] categorized aerodynamic whistles into three classes based on the types of feedback, whose occurrence highly depends on the geometric configuration. A survey of previous studies (with the relevant references given below) on flow-sustained tones in various geometric configurations suggest that jet tone, jet-obstacle interaction, jet-obstacle-resonator interaction, and pipe tone could potentially play a role in tonal SGS sound generation. To facilitate further discussion in the paper, each tone mechanism is briefly introduced here. 1. Jet tone. Circular and planar jets exhibit inherent instability due to the shear layer, causing the axial flow speed to oscillate. The Strouhal number of such oscillations is roughly constant, depending on the orifice geometry, such as nozzles or sharp edges [25; 26; 27]. For axisymmetric ducted jets, the Strouhal number is approximately 0.5 [28; 29]. Small acoustic perturbations can be amplified by the instability of circular jets, leading to shedding vortices in the free boundary layer [29]. Such amplification is selective with the preferred Strouhal number between 0.2 and 1.0 [30; 31]. 2. Jet-obstacle interaction. Jet-obstacle systems consist of an upstream narrowing, such as through an orifice or a slit, that produces a jet, and an obstacle that is closely located downstream. The tones produced in these systems are often named after the shape of the obstacle, with the most common being the edge tone and the hole tone, both of which tend to operate within a relatively low (<3000) Reynolds number range [31]. In such systems, flow-sustained tones can be generated in a series of vortex modes characterized by the number of vortices between the orifice and the obstacle [32]. While distinct vortex shedding is often observed, it is not necessary for the vortex to impinge on the hole plate to generate the tone [14; 33]. Thus, it is more appropriate to regard the vortex as a sign of disturbance than the source of the disturbance in the orifice. As this disturbance is convected downstream, leading to an impingement on the obstacle, it causes a pressure pulse that propagates upstream at sound speed, which is amplified by the jet to form a loop [14; 31]. The SGS and the vocal folds form a potential jet-obstacle system. 3. Jet-obstacle with resonator. The addition of a resonator to the jet-obstacle system causes the frequency of the excited tones to approach the resonance frequencies of the resonator, resulting from the coupling of the vortex mode of the jet-obstacle system and the acoustic mode of the resonator [34; 35; 36]. Resonance with the resonator also allows the jet-obstacle system to operate in a much larger range of Reynolds numbers (up to 200,000 [31]). Wind instruments, such as recorders and ocarinas, are examples of this type of system. An axisymmetric jet-driven Helmholtz resonator (JDHR) [31], which is essentially a tubular cavity enclosed by two concentric orifice plates, is a special configuration in this category. Although higher modes can exist in the axisymmetric JDHR at high Reynolds numbers [37], previous experiments have mainly focused on exciting the fundamental acoustic mode [31; 33; 38] for which the acoustic pressure inside the cavity oscillates uniformly. The cavity enclosed by the SGS and the vocal folds forms a potential Helmholtz resonator. 4. Pipe tone. In pipe tone systems, a jet (which is typically circular) is coupled with a pipe resonator. This system differs from the jet-obstacle system in that the acoustic pressure is generated directly at the first and only orifice [39] without any interaction with an obstacle. SGS in the central airway presents interesting potentials for flow-sustained tones. The narrowing due to the SGS produces a jet flow that is inherently unstable. The vocal folds could function as obstacles, causing acoustic perturbations to the jet flow.