# Study on Band Gap and Sound Insulation Characteristics of an Adjustable Helmholtz Resonator  
[PERSON]  
Correspondence: [EMAIL_ADDRESS] (D.H.); [EMAIL_ADDRESS] (J.Z. );  
Tel. : +86-17-9236-8509 (D.H.); +86-180-8921-8197 (J.Z.) [PERSON]  
Correspondence: [EMAIL_ADDRESS] (D.H.); [EMAIL_ADDRESS] (J.Z. );  
[PERSON]  
Correspondence: [EMAIL_ADDRESS] (D.H.); [EMAIL_ADDRESS] (J.Z. );  
[PERSON]  
Correspondence: [EMAIL_ADDRESS] (D.H.); [EMAIL_ADDRESS] (J.Z. );  
Tel. : +86-17-9236-8509 (D.H.); +86-180-8921-8197 (J.Z.) [PERSON]  
Correspondence: [EMAIL_ADDRESS] (D.H.); [EMAIL_ADDRESS] ([PERSON]. );  
[PERSON]  
Correspondence: [EMAIL_ADDRESS] (D.H.); [EMAIL_ADDRESS] (J.Z. );  
Tel. : +86-17-9236-8509 (D.H.); +86-180-8921-8197 (J.Z.) [PERSON]  
sound focusing, and directional propagation of elastic waves. The typical characteristics of phononic crystals is regulating the band structure of elastic waves, thus obtaining special properties in spectral space, wave vector space and phase space. The most important property is the \"band gap\" characteristic of elastic waves or sound waves, whose feature is as follows: Sound waves or elastic waves within the frequency range of band gap will be suppressed and cannot propagate in the structure, while sound waves and elastic waves outside the frequency range can normally propagate without being affected [10]. The characteristic lays a foundation for its application in the field of noise control. Studies show that there are two main mechanisms for the generation of band gap: Bragg scattering mechanism and local resonance mechanism. Bragg band gap is mainly controlled by Bragg conditions. In order to meet the reflection and stacking effect of elastic waves in periodic structure, its lattice size should be larger than half of the wavelength of elastic waves. Therefore, in order to obtain low-frequency Bragg band gap, the size of the phononic crystal is often too large for practical application. However, locally resonant phononic crystals break through the limitations of Bragg scattering phononic crystals and show obvious advantages in low-frequency control, showing negative equivalent modulus and negative equivalent mass density [11; 12; 13; 14], which are different from traditional materials. Helmholtz resonator is a common locally resonant phononic crystal structure. The common Helmholtz resonator structure is shown in Figure 1, whose principle is based on the air resonance characteristics. The structure has good advantages of light weight and low-frequency design, so it has been recognized by many scholars [15; 16; 17; 18]. [PERSON] [19] proposed an ultra-thin water-based metamaterial composed of multiple Helmholtz resonators, achieving an ideal low-frequency sound absorption effect. [PERSON] [20] designed the Helmholtz structure of a multi-layer open resonant ring, which could adjust the range of low-frequency band gap by changing the opening direction of each layer of the resonant ring. [PERSON] et al. [21] developed a horn-shaped-neck Helmholtz resonator and verified its effectiveness in low-frequency noise reduction through experiments and simulations. There are few researches on low-frequency sound insulation of periodic structure of Helmholtz resonators, especially in the low frequency range below 100 Hz. The main reason is that the length of the tube needs to be increased to improve the low-frequency characteristics, which will reduce the volume of the inner cavity and narrow the band gap. In this paper, the periodic structure of Helmholtz resonator was optimized through two aspects: increasing the length of the tube and changing the cavity structure. A periodic structure of a Helmholtz resonator with adjustable cavity and labyrinth tubes was constructed. Firstly, the structure adopts a double-cavity and double-tube design, which can effectively increase the region of local resonance in the structure based on the local resonance mechanism of multi-cavity coupling [22].