# Leakage Effect on the Transmission Properties of a Duct Loaded with a Helmholtz Resonator
###### Abstract  
The characteristics of transmitted acoustic field have important significance to the leakage detection and the acoustic metasurface technology. When the additional leak holes are present, the conventional single neck Helmholtz resonator will naturally become the one with multiple necks. Based on such a background, in this paper, the effects of leakages on the transmission properties of a Duct Helmholtz Resonator (DHR) device is investigated both analytically and numerically. A set of closed-form formulas are derived to analytically predict the transmission spectra of the DHR device with leakages. The theoretical results are compared with COMSOL predictions. The simulation results show that the number and width of leak holes have significant influences on the amplitude, phase shift of the transmitted wave, and resonance frequency of the DHR system.  
Duct-Helmholtz-Resonator; leakage; acoustic transmission; multi-neck Helmholtz resonator; closed-form formulas +
Footnote †: journal: Applied  
0000-0002-4000-3999-22000-0000-0000-0000-0000-0000-0000-0000-0000-0000-0000-0000-00000-0000-00000-00000-00000-00  
near the resonance, and the transmission loss is almost independent of the locations of the leakage.  
When the leakages are present, the cavity will have one or more additional channels (leak holes) to communicate with the main duct. In this case, the original Helmholtz resonator naturally becomes the one with multiple necks. Up to now, only a few results have been reported on the acoustic characteristics of the multi-neck Helmholtz resonator. [PERSON] et al. (2019) developed an analytical model to calculate the resonance frequency and the input impedance of a Helmholtz resonator with multiple necks [19]. The results show that small additional holes in the Helmholtz resonator have significant influences on the resonance frequency and the absorption performance. [PERSON] et al. (2019) designed a dual-tube Helmholtz resonator-based triboelectric nanogenerator (TENG) for efficient harvesting of acoustic energy [20]. Based on the coupling mechanisms of TENG and acoustic propagation, they provided a theoretical guideline for improving energy output and broadening the frequency band. These limited studies provide a basis for understanding the effect of multiple necks on the acoustic characteristics of Helmholtz resonators.  
Since the transmitted wave carries physical information of the leakage, the acoustic transmission properties of the DHR device with multiple necks are of great significance for the leakage detection and localization of gas pipeline system or metasurface structures [21; 22; 23]. To examine the effects of leakages on the acoustic performance of the DHR device, in this paper, a set of explicit formulas were derived. The effects of the number and the size of leak holes, and the distance between the leak hole and the main neck on the transmission properties of DHR were predicted. Compared with the COMSOL finite element (FE) simulations, the developed theoretical formulas can be employed to better understand and quickly estimate the impact of leakage on the transmission properties of the DHR device, and thus have significant meaning in the field of oil and gas pipelines, as well as acoustic metasurface engineering.