Acoustic Properties of Surfaces Covered by Multipole Resonators

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

This paper continues a series of previous papers by the author and is closely linked to Ref.13 published in Acoustics in 2022.

The authors reformulate expression for obliquely incident plane wave reflection coefficient by introducing a so called tangential impedance proportional to the ratio of the order 2n pressure derivative with respect to the horizontal spatial variable  to the normal velocity component.

The paper is very similar to Ref.13 and I am not convinced that there is enough new material present in it to warrant a publication.

My suggestions for the improvements in addition to those given in annotated pdf would be the following:

1.      Give examples. Multipole resonances normally have high frequencies, hence short wavelengths. Satisfying the condition of the distance between the scatterers on the surface being smaller than the wavelength might be tricky. Show how this can be done by considering several examples. An example of a membrane resonator has been considered in Ref.13. Can something similar be done for this paper too?

2.      For these examples, relate tangential impedance given by equation (29) to the geometrical and physical parameters of the multipole resonators and the distances between them on the surface.

3.      Plot the graphs as in Figs. 3-5 for these examples. Otherwise, it is not clear how realistic the parameters considered for the existing Figs. 3-5 are.

4.      The credibility of the results would be improved if comparisons with numerical models such as FEM or experimental results are present, as was done e.g. in Ref. 12.

5.      Consider the application of homogenisation model as e.g. in Marigo & Maurel JASA, 2016 to derive the higher order equivalent boundary conditions.

Comments for author File: Comments.pdf

Comments on the Quality of English Language

needs to be improved but understandable

Author Response

Dear Reviewer,

I am very grateful for your attention to this manuscript and your opinion concerning the previous paper. You are right, the manuscript develops the concept of the tangential impedance in relation to metasurfaces with complex structures. The previous study focused on surfaces covered by dipole resonators and identified a homogenized boundary condition. This paper extends this research to resonators of any order and multiplicity, and the equivalent boundary conditions are strongly dependent on the order of the multipole and can be used to describe the acoustic properties of surfaces formed by these resonators. In my opinion it is an important example of appearing of high order impedance boundary conditions in the field of acoustic metasurfaces.

The manuscript has been revised taking into account your suggestions including remarks shown in the pdf file. Below you can find comments to your remarks:

 

1. Give examples. Multipole resonances normally have high frequencies, hence short wavelengths. Satisfying the condition of the distance between the scatterers on the surface being smaller than the wavelength might be tricky. Show how this can be done by considering several examples. An example of a membrane resonator has been considered in Ref.13. Can something similar be done for this paper too?

Thank you for the suggestions. The proposed theory considers multipole resonators in general view. They are described by the impedance. To use it we need a multipole moment, which depends on a construction of the resonator. It is a good idea to demonstrate how the multipole moment can be found for specific systems of the resonators. After finding the moment the proposed theory can be applied directly.

Following the previous work, I have analyzed sets of Helmholtz resonators and membranes resonators and I have added these findings in the Section “Discussion”. I hope that readers can now imagine the real multipole resonator.

2. For these examples, relate tangential impedance given by equation (29) to the geometrical and physical parameters of the multipole resonators and the distances between them on the surface.

Equation (29) works for introduced examples of the Helmholtz resonators and membranes resonators if the distance between the resonators is small in comparison with their lengths, i.e. .

3. Plot the graphs as in Figs. 3-5 for these examples. Otherwise, it is not clear how realistic the parameters considered for the existing Figs. 3-5 are.

Figures 3-5 are valid for any kind of resonators if equation (29) can be applied i.e. if . The examples of the resonators considered in the section “Discussion” can be used to construct the metasurface. I focus on the particular case of the small resonators, but if the condition  is not satisfied, the problem should be considered separately. I guess that if  the effect of the multipole scattering would be blurred, but this point requires another study.

4. The credibility of the results would be improved if comparisons with numerical models such as FEM or experimental results are present, as was done e.g. in Ref. 12.

This paper presents an analytical approach to the problem and provides the theoretical basis for surfaces covered by multipole resonators. The experiment is an essential stage of the current study. I hope to carry out the experiments with dipole and quadrupole resonators, as well as arrays made of them. However, today we are not close to start the experiment. Regarding the simulation, I believe it will be very beneficial for the resonant systems with finite sizes.

5. Consider the application of homogenisation model as e.g. in Marigo & Maurel JASA, 2016 to derive the higher order equivalent boundary conditions.

Thank you for this reference. I think it proposes a very useful approach for rigid surfaces. The considered resonators are not rigid, so the method can not be directly applied for the resonant surfaces. However, I mentioned it as an example of the deriving of higher order equivalent boundary conditions.

Reviewer 2 Report

Comments and Suggestions for Authors

The manuscript investigates the acoustic properties of metasurfaces composed of multiple resonators, specifically focusing on the theory behind acoustic metasurfaces, sound reflection, and absorption properties. It employs a theoretical model to elucidate how the acoustic properties of metasurfaces can be homogenized using a high-order tangential impedance boundary condition, depending on the multipole order of the resonators. This work is interesting and generally well-organized. Mathematical analysis work is worth encouraging in the current environment where simulations are rampant. I recommend this manuscript be accepted after minor reversions.

 1. Can the author demonstrate specific examples? It would help readers understand more intuitively.

2. The manuscript discusses the relationship between scattered fields and the distance between the scatterers. I think the concept of “subwavelength” is related and deserves to be mentioned.

3. Typos needs to be corrected.

Author Response

Dear Reviewer,

I would like to thank you for your feedback and support regarding the theoretical approach to this issue. The manuscript has been significantly revised taking into account your remarks.

1. Can the author demonstrate specific examples? It would help readers understand more intuitively.

Some examples are given now in the Section “Discussion”

2. The manuscript discusses the relationship between scattered fields and the distance between the scatterers. I think the concept of “subwavelength” is related and deserves to be mentioned.

I agree with you the concept of “subwavelength” was actually used. I added some mentions about it.

3. Typos needs to be corrected.

Some typos were found and corrected.

Reviewer 3 Report

Comments and Suggestions for Authors

My comments are in the file

Comments for author File: Comments.pdf

Comments on the Quality of English Language

My comments are in the file

Author Response

Please, see the attachment

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

Thank you for addressing my comments. My recommendation now is publication after minor revision. Some suggestions for improvement are in the attached pdf with annotations highlighted in pink.

Comments for author File: Comments.pdf

Comments on the Quality of English Language

I corrected some of the obvious errors in the attached pdf

Author Response

Dear Reviewer, 

I am very grateful for your recommendations. They are very helpful and valuable.

All changes in the revised manuscript are shown in green.

With kind regards,

Nikolay Kanev