In order to verify the sound absorption effect of Helmholtz cavity at the resonant frequency, the commercially software, COMSOL Multiphysics 5.5, was used in the analyses.

The plane wave radiation is applied on one side and transmission loss of sound propagation is measured on the other side. The schematic diagram is shown in Fig. 3. Transmission loss is defined bellow:

$$TL{\text{=}}{L_i} - {L_t}=10\lg \frac{{{I_i}}}{{{I_t}}}$$

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Where\({L_i}\) and \({L_t}\) are the power of incident sound and transmission sound, and \({I_i}\),\({I_t}\) are the power of the incident sound and transmission sound, which are defined as follows:

$$\begin{gathered} {I_i}=\int_{{{S_P}}} {\frac{{{{\left| {{P_i}} \right|}^2}}}{{2\rho {c_0}}}ds} \hfill \\ {I_t}=\int_{{{S_P}}} {\frac{{{{\left| {{P_t}} \right|}^2}}}{{2\rho {c_0}}}ds} \hfill \\ \end{gathered}$$

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Where\({P_i}\) and \({P_t}\) are the pressure of incident sound and transmission sound, \({S_P}\) is the area of the inlet and outlet. The transmission coefficient of the sound intensity is defined below:

$$T=\frac{{{I_t}}}{{{I_i}}}=\frac{{{{\left| {{P_t}} \right|}^2}/2\rho {c_0}}}{{{{\left| {{P_i}} \right|}^2}/2\rho {c_0}}}$$

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If the inlet and outlet areas of the pipeline are the same, the relationship between the loss and transmission coefficient can be defined as follows:

$$TL=10\lg \left( {{t^{ - 1}}} \right)$$

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The transmission loss is calculated using COMSOL, which is shown in Fig. 4. The muffling frequency is around\(480{\text{Hz}}\), which has a good agreement with the resonant frequency, also, the transmission loss reaches nearly 45dB. It is proved that the Helmholtz cavity can truly have good silencing properties near its resonant frequency.

However, if we want to get a transmission loss over 30dB, the range of absorption frequency is only 12.12Hz. The reason is Helmholtz cavity can only get a very narrow muffling peak in its resonant frequency. Obviously, the result does not satisfy the need of broadband noise reduction. Through the derivation of the theory in the previous section, it can be concluded that if more different resonance frequencies can be obtained, and these frequencies are close enough, the range of frequency can definitely increases.

In this paper, we propose the concept of a gradient resonance frequency, the resonant frequency position is controlled by adjusting the geometric parameters inside the Helmholtz cavity, which are defined as follows:

$$\begin{gathered} {{\text{D}}_n}={{\text{D}}_1}\cdot {\text{(0}}{\text{.9+0}}{\text{.1n) }} \hfill \\ {{\text{h}}_n}={{\text{h}}_1}\cdot {\text{(0}}{\text{.9+0}}{\text{.1n) }} \hfill \\ \end{gathered}$$

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Where \({{\text{D}}_n}\)and \({{\text{h}}_n}\)are the diameter and height of cylindrical cavity, n represents the number of Helmholtz cavities, and\({{\text{D}}_1}{\text{=}}8{\text{cm}}, {{\text{h}}_1}{\text{=}}3{\text{cm}}\).

The first case to be considered is n = 2, which is depicted in Fig. 5(a). The transmission loss is also calculated using COMSOL. As depicted in Fig. 5(b), it can been seen that there are two distinct resonance frequency peaks. Also, the range of frequency increases to 112.13Hz when the transmission loss is over 30dB. That's a tenfold increase compared to n = 1. So, the width of reduction frequency can be greatly broadened by arranging Helmholtz cavities in a gradient way.

To make the results more obviously, three cases n = 4, 6, and 8 are considered, respectively. Based on the transmission loss of 30 dB, the frequency range of sound absorption in different numbers of cavities was measured. In Fig. 6(a), four gradient cavities are added in the pipe, result is shown in Fig. 6(b), width of the frequency to reduce noise reach up to 203.28Hz. Results show that the width of silencing frequency increases as the number of cavities increases, and the transmission loss at the peak is increasing. The same phenomenon appears in six cavities and eight cavities. The schematic diagram is shown in Fig. 7(a) and Fig. 7(c) and transmission loss is shown in Fig. 7(b) and Fig. 7(d), the width can reach up to 230.86Hz and 273.54Hz, almost 22.5 times that of one cavity. This result is quite striking and perfect. This is a big improvement compared to only one Helmholtz cavity.

The Helmholtz cavity theory is used to reduce the noise of the structure. Helmholtz cavity structure is composed of a large cavity and a small neck. Since the volume of the neck is much smaller than that of the cavity, the fluid in the neck can be considered approximated incompressible. However, the fluid in the neck is considered compressible under the motion of the neck fluid. As the fluid in the cavity is compressed, the pressure increases naturally, providing a reverse restoring force. According to the previous description, the fluid in the neck can be regarded as a mass and the fluid in the lumen can be regarded as a spring. On this basis, in order to increase the number of resonance frequency of the entire structure, we adjust the geometrical parameters of the cavities so that they can varies with gradient. When the excitation frequency is close to the resonant frequency, the Helmholtz cavities can absorb sound waves. Figure 8 shows the acoustic pressure of this muffler in frequency f = 400Hz when there is four Helmholtz cavities, the result shows that most of the sound pressure in the pipe is absorbed into the cavity. Because the gradient variation of Helmholtz cavity has affected the resonance frequency, the original resonant frequencies is overlapped. Although 400Hz is not a resonant frequency, the whole structure can still achieve a good sound attenuation performance at 400Hz

When noise enters the muffler, the Helmholtz cavities act as a filter, preventing sound waves of a certain width of frequency from passing through. However, when there is only one or two cavities, sound waves cannot be fully absorbed, and some will escape from the Helmholtz cavity. By comparing different cases, it is easy to find the width of the absorption frequency increases by the number of cavities gradient-increasing, and as the number increases, the lifting speed gradually slows down, eventually close to 300Hz, almost 24.75times that of one cavity, which is shown in Fig. 9. In addition, the transmission loss around the center frequency also increases from 45dB to near 100dB. So, as the number of cavities increases, not only the range of band gap is broadened, but also the ability to absorb sound waves is improved.

In order to further study the relationship between noise attenuation characteristics and the distance of the cavities, another simulation is made by changing their distance from 15 cm to 37.5 cm. We can notice from the transmission loss in Fig. 10. (a), a valley gradually appears at 420Hz, and the valley becomes more and more obvious when the distance of the cavities increases. The valley splits the formerly wide muffler band in two, makes the resonant frequencies of the two cavities independent of each other. The phenomenon results in the structure not being able to absorb sound in wide range of silence frequency, and reduces the muffler ability to some extent. Figure 10(b) and Fig. 10 (c) shows the acoustic pressure of two cavities with the distance 15cm and 37.5cm, respectively. The frequency of 420Hz at the valley is selected, it is easy to find that most of the sound pressure in the pipe is absorbed into the cavity when the distance is 15cm, prove that the valley has little effect on the noise-reduction feature. However, when the distance is 37.5cm, most sound pressure appears in the end of the pipe, as a result, the Helmholtz cavity basically no longer has sound absorption ability in the valley.