# Study on Band Gap and Sound Insulation Characteristics of an Adjustable Helmholtz Resonator
## 3 Modeling and Analysis of Low-Frequency Band Gap Formation Mechanism  
### Study on Band Gap Mechanism  
In order to explore the mechanism of generation of band gaps and analyze the characteristics of low-frequency sound insulation, the sound pressure fields and vibration modes of point A, B, C and D in Figure 6 were analyzed in detail. Points A, B, C and D represent the upper and lower limits of the first and second band gaps respectively. The relationship between band gap frequency and cavity vibration was revealed by analyzing the band diagram and the eigenvibration modes of cavity II, cavity I and the outer cavity. The sound pressure fields of the structure at point A, B, C and D are shown in the Figure 7.  
Figure 7a shows the sound pressure field at point A. As shown in Figure 6, the frequency at point A is the maximum of the first band curve, so point A is the lower limit of the first complete band gap. It can be found from the sound pressure diagram that the distribution of sound pressure presents regional uniformity, which proves that the resonance state appears at point A. Meanwhile, as shown in Figure 7a, the sound pressure of the cavity I and the outer cavity is almost 0, so the sound pressure is almost localized inside the cavity II. In addition, the sound pressure in the labyrinth tube of cavity II varies in a gradient. The amplitude of the sound pressure is large in the region close to the cavity II, and small in the region close to the outer cavity, indicating that the local state appears at  
Figure 5: (**a**) Transmission spectrum of the phononic crystal; (**b**) transmission spectrum of the control group. Figure 6: Band diagram of the structure. point A. The vibration mode of the phononic crystal at the first lower limit is identical to the eigenvibration mode of the cavity II. Under the combined action of resonance state and local state, the first band at point A becomes a straight band. At the same time, the sound waves are localized in the cavity II, so the first band gap appears. Figure 7b shows the sound pressure field at point B. It can be seen from the figure that the sound pressure field at point B is significantly different from point A. At this point, the amplitude of the sound pressure in the outer cavity is smaller than that in the inner cavity, and the phase of the sound pressure in the cavity II is opposite to that in the cavity I. It shows that the sound waves can propagate outside and inside the structure, and the structure cannot prevent the sound waves from propagating through the structure. Therefore, point B determines the cutoff frequency of the first band gap. As shown in Figure 7c, the sound pressure field at point C is similar to that at point A, which also presents a local resonance state. Different from point A, the vibration of the sound pressure field at point C is localized to the cavity I, and the sound pressure of the cavity II and the outer cavity is very small, almost 0. In the labyrinth tube of the cavity I, the sound pressure varies in a gradient and gradually decreases from the inside to the outside. The vibration mode of the phononic crystal at the first lower limit is identical to the eigenvibration mode of the cavity I. This indicates that the sound waves are localized to the cavity I, and at this time, the second band gap appears. Figure 7d shows the sound pressure field at point D. At point D, the sound pressure is distributed inside and outside the structure, and the sound pressure of outer cavity is higher than the inner cavity. The phase of the sound pressure in cavity II is the same as that in cavity I, which means that sound waves can propagate through the structure without  
Figure 7: The sound pressure distribution diagram of point A (**a**), point B (**b**), point C (**c**) and point D (**d**). being affected. Therefore, point D determines the cutoff frequency and the upper limit of the second band gap.