Leakage Effect on the Transmission Properties of a Duct Loaded with a Helmholtz Resonator
Round 1
Reviewer 1 Report
The authors investigate theoretically the acoustic wave transmission through a duct Helmholtz resonator containing additional holes. They derive analytic formulas for the transmission coefficient and, in particular, show that the above-mentioned extra holes always shift the resonance to higher frequencies. Analytical results are compared with the result of finite element simulations by Comsol.
I think the paper can be published after some revision.
Namely, please indicate clearly the novelty of the theory developed in this paper (section 2) as compared to that of Refs. [17,19].
Author Response
Comment: Please indicate clearly the novelty of the theory developed in this paper (section 2) as compared to that of Refs. [17,19].
Response: Special thanks to you for your good comments. The theory in Refs. [17, 19] calculated the impedance and resonance frequencies of multiple necks as a whole. The theory developed in this paper calculated the impedance at the intersection of each neck and pipe separately in section 2.1, and the equivalent impedance of the whole structure is obtained by transfer matrix method in section 2.2. The theory in this paper can be used to calculate the amplitude and phase shift of acoustic wave passing through the DHR device and the results are in good agreement with the finite element simulation.
Reviewer 2 Report
The paper presents the development of a theoretical model for duct Helmholtz resonator (DHR as referred to in the text). The model is then validated numerically via COMSOL. Following the validation, parametric study was performed to understand the characteristics of various dimensions.
It is unclear how the theoretical model developed distinguish itself from past works (doi: 10.1121/1.3672692 is one of the many examples). The paper claims to investigate the leakage effect but this effect appears to be simply the resonance response of the DHR due to different dimensions (which is not novel).
Moreover, the performance are mainly targeted at narrowband high-frequencies, which may have limited practical use. It would be good to consider how such DHR can potentially be applied.
In summary, the scientific quality of this paper cannot meet that of Applied Sciences and would recommend a rejection.
Author Response
- Comment: It is unclear how the theoretical model developed distinguish itself from past works (doi: 10.1121/1.3672692 is one of the many examples). The paper claims to investigate the leakage effect but this effect appears to be simply the resonance response of the DHR due to different dimensions (which is not novel).
Response: Special thanks to you for your good comments. Most previous studies on DHR devices focused on single-neck condition, with one neck for each cavity. The theory does not apply to the case that a single cavity contains multiple necks. A few studies on multiple necks calculated the impedance and resonance frequencies of multiple necks as a whole, which are inaccurate in the calculation of transmission properties. The method developed in this paper predicted the impedance at the intersection of each neck and pipe separately in section 2.1, and the equivalent impedance of the whole structure is obtained by transfer matrix method in section 2.2. The theory in this paper can be used to calculate the amplitude and phase shift of acoustic wave passing through the DHR device and the results are in good agreement with the finite element simulation.
- Comment: The performance are mainly targeted at narrowband high-frequencies, which may have limited practical use. It would be good to consider how such DHR can potentially be applied.
Response: Special thanks to you for your good comments. The width of the high-transmittance band can be increased by paralleling more cavities along the direction of the pipe. In addition to the resonance frequency, the analytical method established in this paper can also quickly predict the amplitude and the phase characteristics of the transmission, which is of great significance for the leakage detection and localization of the gas pipeline system and DHR-based metasurface applications. By paralleling DHR devices with different transmission phase shifts, the transmitted wavefront can be controlled, so that anomalous refraction, acoustic focusing, and other functions can be realized.
Reviewer 3 Report
In the submitted manuscript, the authors develop a 2D analytical model of a multiple-neck Helmholtz resonator attached to a waveguide, and employing this model, the transmission properties of several configurations are discussed. The problem of reduction of sound propagating in guides is an important engineering task which has received a lot of attention in the past.
Even if the authors’ model is rather complex (and what is more, it describes only a quite unrealistic 2D geometry) and today, tools such as finite elements are accessible and they allow fast and simple study of these (and more realistic) systems, the developed model may be of interest, however, the authors have to show its benefits.
Even if the authors compare their results with the ones by finite elements, they do not show a comparison of their results with the ones which would be obtained employing a simpler model, for example, of an equivalent lumped-element Helmholtz resonator with one neck whose width equals the sum of the widths of all the necks. Or, what would be the difference between the numerical results, if there were two necks, whose mutual distance changes?
If these issues are not addressed, it is not clear what the benefit of the proposed approach is.
Specific issues:
- Line 99: Base on -> based on
- Line 326: The authors study the properties of one ducted Helmholtz resonators. How can this be modelled employing periodic boundary conditions?
- Figure 5 and elsewhere: acoustic impedance is not a dimensionless quantity.
- Line 342: The authors do not mention any model of acoustic energy absorption, and yet, they talk about “perfect absorption.” They should be aware of the fact that if no acoustic energy absorption is introduced in the model, the zero transmission is caused by the fact that due to the impedance mismatch, all the acoustic energy is reflected (and not absorbed).
- Line 348: the part of the sentence “the transmitted wavefront of the acoustic metamaterial” does not make any sense.
- Figure 7: What is shown in this figure is rather trivial and it is probably not worthy being presented in a scientific journal.
Author Response
- Comment: The problem of reduction of sound propagating in guides is an important engineering task which has received a lot of attention in the past. Even if the authors model is rather complex (and what is more, it describes only a quite unrealistic 2D geometry) and today, tools such as finite elements are accessible and they allow fast and simple study of these (and more realistic) systems, the developed model may be of interest, however, the authors have to show its benefits.
Response: 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, which can be used in the design of DHR device easily and quickly.
- Comment: Even if the authors compare their results with the ones by finite elements, they do not show a comparison of their results with the ones which would be obtained employing a simpler model, for example, of an equivalent lumped-element Helmholtz resonator with one neck whose width equals the sum of the widths of all the necks. Or, what would be the difference between the numerical results, if there were two necks, whose mutual distance changes? If these issues are not addressed, it is not clear what the benefit of the proposed approach is.
Response: The comparison between the different neck distances and the equivalence of the two necks to a wider neck is supplemented in Fig. 7. The results show that the amplitude and phase shift of the DHR device are influenced by the distance of necks, and it is not equivalent to a neck whose width is equal to the sum of several neck widths. From Appendix A, we can also see the distance between the necks affects the value of the transfer matrix and finally affects the amplitude and phase shift. So, the theory in the manuscript can be used to calculate the transmission properties of DHR device in multi-neck condition more accurately comparing with traditional methods.
- Comment: The authors study the properties of one ducted Helmholtz resonators. How can this be modelled employing periodic boundary conditions?
Response: Floquet periodic boundary conditions are applied to the upper and lower boundaries of the incident and transmitted wave regions in Fig. 3 to simulate the continuous arrangement of multiple DHR devices.
- Response to comment: Figure 5 and elsewhere: acoustic impedance is not a dimensionless quantity.
Response: We are sorry for our negligence of the unit of the impedance, the unit is added in themanuscript.
- Comment: Line 342: The authors do not mention any model of acoustic energy absorption, and yet, they talk about perfect absorption. They should be aware of the fact that if no acoustic energy absorption is introduced in the model, the zero transmission is caused by the fact that due to the impedance mismatch, all the acoustic energy is reflected (and not absorbed).
Response: We are very sorry for our mistake. The corresponding part in the manuscript is changed to perfect reflection.
- Comment: Line 348: the part of the sentence the transmitted wavefront of the acoustic metamaterial does not make any sense. Figure 7: What is shown in this figure is rather trivialand it is probably not worthy being presented in a scientific journal.
Response: Special thanks to you for your good comments. These two partsare deleted in the manuscript.
Round 2
Reviewer 2 Report
Despite the response provided, the revised paper did not fully differentiate the theoretical model developed distinguish itself from past works, which are highly similar.
The authors did not provide any response to the original comment in which the leakage effect appears to be simply the resonance response of the DHR due to different dimensions (which is not novel).
Hence, at this point, the novelty of this paper remains unclear and lacking. The authors are welcomed to improve further.
Author Response
Comment: Despite the response provided, the revised paper did not fully differentiate the theoretical model developed distinguish itself from past works, which are highly similar. The authors did not provide any response to the original comment in which the leakage effect appears to be simply the resonance response of the DHR due to different dimensions (which is not novel). Hence, at this point, the novelty of this paper remains unclear and lacking.
Response: Special thanks to you for your good comments. Up to now, there is no theoritical formulas on calculating the transmission characteristics of the multi-neck Helmholtz resonator. Compared with past works, the main difference of the proposed theory in this manuscript is that multiple necks are connected in the same cavity and the influence of the distance between the necks on transmission characteristics is considered. Although the resonance characteristics depend mainly on the size of the neck and cavity, the distance between the necks also has influence on the resonance frequency. As shown in the modified Fig. 7, although in traditional theory, the distance between the necks has no effect on the resonance frequency, both the theory in this paper and the simulation results do not accord with this conclusion. Therefore, compared with the traditional theory, the proposed theory in this manuscript can better predict the transmission characteristics on multi-neck condition.
Reviewer 3 Report
It is fact that the authors have included Fig. 7, where they present the change of the transmission properties depending on the distance between the holes, but, unlike all the other figures, they show the results in a quite unineresting frequency range, where the transmission coefficient is close to 1 and the change of the distance between the holes changes the transmission coefficient by one percent or so. Do the authors want to say that their method is capable of increasing the accuracy of the prediction by a few percents compared to the lumped element model? Please, provide a figure where the resonance takes place (such as in Fig. 4).
Author Response
Comment: It is fact that the authors have included Fig. 7, where they present the change of the transmission properties depending on the distance between the holes, but, unlike all the other figures, they show the results in a quite unineresting frequency range, where the transmission coefficient is close to 1 and the change of the distance between the holes changes the transmission coefficient by one percent or so. Do the authors want to say that their method is capable of increasing the accuracy of the prediction by a few percents compared to the lumped element model? Please, provide a figure where the resonance takes place (such as in Fig. 4).
Response: Special thanks to you for your good comments. We are sorry for choosing an unimportant frequency range and have modified Figure 7 to the full frequency range. The results show that the amplitude and phase shift of the DHR device are influenced by the distance of necks, and it is not equivalent to a neck whose width is equal to the sum of several neck widths. Although in traditional theory, the distance between the necks has no effect on the resonance frequency, both the theory in this paper and the simulation results do not accord with this conclusion. As the distances between the necks become coalescent neck, 7mm and 14mm, the resonance frequencies become 1878Hz, 2224Hz and 2396Hz, respectively, which is a big difference that cannot be ignored.
