# Development of a Metasilencer Considering Flow in HVAC Systems
## 2 Theory
### Metasilencer Design  
Based on the acoustic measurement results, the silencer was set to reduce the noise in the frequency range of 800-1100 Hz. The metasurface used herein is characterized by a bandgap reduction of approximately \(\pm\)100 Hz with respect to the target frequency. Therefore, the metasurface target frequencies were set to 900 Hz and 1000 Hz. Figure 4a shows the metasilencer duct model, where the upper and lower parts of the metasilencer reduce noise at 900 Hz and the width is set to 90 mm, while the left and right sides reduce noise at 1000 Hz and the width is set to 60 mm. In Figure 4a, \(c\) is the width of the metasurface and \(d\) is the height of the metasurface. The maximum width of the metasurface was set to 200 mm, and the maximum height was set to 45 mm. Figure 4b shows the shape of the unit cell structure, and the structure of the unit cell is set to a labyrinthine type. The reason for having a labyrinthine-type structure is that the target frequency can be designed at a lower height than that of the same geodesic tube, which is advantageous in terms of space constraints, where \(g\), \(e\), \(h\), and \(l_{n}\) denote the width of the unit cell, width of the unit cell cross section, wall thickness, and length of the unit cell entrance, respectively. The width of the unit cell (\(g\)) was set to 21 mm and 19 mm according to the target frequency of 900 Hz and 1000 Hz, respectively; the width of the unit cell (\(e\)) and wall thickness (\(h\)) cross sections were set to 3 mm and 1 mm, respectively, regardless of the target frequency. The metasilencer was designed by adjusting the unit cell interval (\(dx\)) and unit cell entrance length (\(l_{n}\)) to prevent the sound wave propagation direction of the target frequency in a U-shaped configuration. In other words, the factors for designing the acoustic metasilencer are the unit cell spacing and length of the unit cell entrance. The unit cell interval can be obtained using the generalized [PERSON]'s law, and the length of the unit cell entrance can be obtained using the phase delay. First, the generalized [PERSON]'s law is  
\[sin\theta_{r}=sin\theta_{i}+\frac{\lambda}{2\pi}\times\frac{d\varpi}{dx} \tag{1}\]  
Figure 4: 3D model: (**a**) metasilencer duct; (**b**) unit cell. Figure 3: 3D model with straight pipe in simple model. where \(\theta_{i}\), \(\theta_{r}\), \(\lambda\), \(d\varnothing/dx\), \(dx\), and \(d\varnothing\) denote the angle of incidence, angle of reflection, wavelength of the target frequency, phase gradient, unit cell spacing, and phase delay interval, respectively. The angle of incidence and angle of reflection can be set to refract in the desired direction, and the length of the wavelength can be obtained by determining the target frequency. In the generalized Snell's law, the variable is the phase gradient, and setting the interval of the phase delay to a constant value in the phase gradient can induce the spacing of the unit cell. Figure 5 shows how the angles of incidence and reflection can be set, where the vertical part of the surface of the metasurface is \(0^{\circ}\), and the direction of sound wave propagation is \(-90^{\circ}\). For example, if the incident angle and reflection angle are set to \(30^{\circ}\) and \(60^{\circ}\) on the metasurface, respectively, the incident angle is \(-120^{\circ}\), and the reflection angle is \(-30^{\circ}\). To refract the traveling direction of the sound wave in a U-shaped configuration, the incident and reflection angles were set to \(-90^{\circ}\) and \(0^{\circ}\) for the upper portion, and the incident angle and reflection angle were set to \(0^{\circ}\) and \(90^{\circ}\) for the lower portion, respectively. The wavelength length of the target frequency can be obtained by \(\lambda=c/f\), where \(c\) is \(343\) m/s because the sound propagation medium is air. The phase delay interval is set to \(30^{\circ}\), and the phase delay is discussed in detail in the next paragraph.