# Study on Band Gap and Sound Insulation Characteristics of an Adjustable Helmholtz Resonator
## 2 Structural Design and Band Gap Analysis
### Band Structure Analysis  
As can be seen from Figure 2, the structure contains two regions: air domain and solid domain. The pressure acoustics module and solid mechanics module were selected in  
Figure 2: Cross section of the structure. Comsol Multiphysics. Since the air had only volume modulus but no shear modulus, the passive acoustic wave equation was needed to calculate its dynamic characteristics:  
\[\frac{\ddot{p}}{c^{2}}=\
abla^{2}p \tag{1}\]  
where \(p\) is the sound pressure value and \(c\) is the sound velocity in the air. For the solid domain composed of the steel, the elastic dynamic equation should be used to calculate its vibration characteristics:  
\[(\lambda+2\mu)\quad\
abla^{2}\mathrm{u}+(\lambda+\mu)\
abla\times(\
abla+ \mathrm{u})=\rho\frac{\partial^{2}\mathrm{u}}{\partial t^{2}} \tag{2}\]  
where, \(\lambda\) and \(\mu\) are [PERSON]'s constants, \(\rho\) is solid density and \(\mathrm{u}\) is the displacement vector. The relationship between the Ramet's constants and the elastic modulus \(E\) and [PERSON]'s ratio \(\
u\) is as follows:  
\[\lambda=\frac{E\
u}{(1+\
u)(1-2\
u)} \tag{3}\]  
\[\mu=\frac{E}{2(1+\
u)} \tag{4}\]  
According to the material parameters of air, we can calculate that the acoustic impedance of air is 428 N\(\cdot\)s/m\({}^{3}\), while for the steel which makes up the structure of Helmholtz resonator, its acoustic impedance is 4.05 \(\times\) 10\({}^{7}\) N\(\cdot\)s/m\({}^{3}\). By comparison, the acoustic impedance of steel is much larger than air, which indicates that when the sound waves propagate from air into the structure, only a small part of the sound energy will enter the structure through the interface between the two, and most of the sound energy will be reflected back into the air. Therefore, the boundary between the two is set as a rigid boundary and the structure can be regarded as a rigid body in the simulation calculation, and the effect of interface phenomena between air and the structure was ignored, thus simplifying the simulation calculation model. When calculating its band structure in Comsol Multiphysics, because the structure is periodic in \(x\) direction and \(y\) direction, according to Bloch theory, two pairs of Bloch-Floquet boundaries were imposed in \(x\) direction and \(y\) direction of the cell to simplify the periodic structure into a single cell structure. The corresponding band gap frequency can be obtained by calculating the characteristic frequency of the cell under this periodic condition. The periodic boundary can be expressed as:  
\[p(r+a)=p(r)e^{ika} \tag{5}\]  
where \(r\) is the position vector, \(a\) is the lattice constant and \(k\) is the wave vector. In order to explore the band gaps and local resonance modes by FEM, the triangular mesh was selected to divide the cell structure, and the finite element meshing is shown in Figure 3. In the finite element simulation experiment, the structure with the parameters as shown in Table 1 was taken as the research object, and simulation calculation was carried out by Comsol Multiphysics. The calculated band structure and transmission spectrum of the structure are shown in Figure 4.  
As shown in Figure 4a, the structure has two complete band gaps (the gray part in Figure 4a in the range of 0-500 Hz. Among them, the first low-frequency band gap ranges from 39.27 Hz to 86.18 Hz, and the second complete band gap ranges from 133.12 Hz to 209.44 Hz. In order to verify the correctness of the calculation of band structure, the transmission spectrum of the periodic structure was made for comparison. In COMSOL Multiphysics, Equation (6) was used to calculate the transmission loss of the sound waves in the range of 0-500 Hz in the periodic structure. As shown in Figure 4b, the suppression range is in good agreement with the band gap, which verifies the correctness of the band gap calculation. \[T=-20\lg\frac{|P_{2}|}{|P_{1}|} \tag{6}\]  
where \(T\) represents transmission loss, \(P_{2}\) represents transmitted sound pressure and \(P_{1}\) represents incident sound pressure.