Diffraction of Sound Waves by a Periodic Array of Impedance Screens

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors The manuscript submitted by Nikolay Kanev to Acoustics Journal (MDPI) represents an interesting contribution in the study of sound diffraction by a periodic louvered barrier with thin impedance screens. The research topic has been previously addressed by the researcher, and other researchers in the context of electromagnetic waves, but the main novelty of the present work is the analytical study of the diffraction problem for an infinite periodic array of impedance thin screens proposed by the author.    The paper is clear and very well written (correct English language is used and no typos/mistakes were found) and well organized, with a classical compact paper structure, with an Introduction, a Materials and Methods section, followed by the Results analyses and a Discussion final section. The illustrative figures and the plots for presentation of results are of good quality and easy to understand. The references introduced by the authors are globally adequate, pertinent and briefly commented or used in the text; they are globally well introduced in the state-of-the-art description or as a justification of the adopted methodologies for the problem analysis. The Abstract is well written and describes the contents of the paper.   In the Introduction, the authors present a good literature review of the focused problem, describing the application of ventilated noise barriers, its features and evolution towards the more recent context of acoustic metamaterials and metasurfaces. In Section 2, firstly, the formulation of the wave diffraction problem is presented, followed by the derivation of the analytical solution and the presentation of the infinite systems of equations that need to be solved, numerically, to evaluate the scattered wave field by the periodic array of impedance thin screens. Section 3 corresponds to the presentation of results, their analyses and discussion, for the cases of screens with arbitrary impedance, non-dissipative screens and dissipative screens, with varying impedance values and a small set of possible geometrical configurations (with changes in screen periodicity and length). The presented results correspond to reflection and transmission coefficients, to transmission loss and absorption coefficient for the periodic array of screens. In the reviewer's opinion, the presented Discussion and conclusions are not very explanatory, resulting from the factual observation of the plots. It could be improved with physical interpretation of the results and useful guidelines towards the practical design of the presented type of louvered noise barriers.   Some typos/errors that the authors should address and or correct: - on page 2, in line 77, check if "[34, 36]" should be corrected to "[35, 36]"; - on page 10, in line 266, check if the reference equation (16) is correct.   Finally, given the global quality of the submitted paper, in the reviewer's opinion, the submitted paper manuscript can be recommended for publication in Acoustics journal in present form after the minor suggestions above mentioned.

Author Response

Dear Reviewer,

First of all, I would like to warmly thank you for the time you spent on the manuscript, and for your very helpful comments.

The main goal of this paper is to investigate how the acoustic impedance of screens affects the incident sound field. The key result is an analytical solution that provides a calculation for arbitrary impedance values. In my opinion, two practical design implications are the most important:

• To minimize the transmitted sound, we need absorptive materials with an impedance value close to 1. Therefore, we can use common porous materials such as wool panels, foams, and others for our design;
• To maximize the reflected sound field, we need non-dissipative screens. These screens can be made from an array of resonators. At a certain frequency range, the homogenized impedance has a negative imaginary part and a zero real part.

These reflections on practical implementation have been incorporated into the revised manuscript. Errors with reference and equation numbers have also been corrected.

Reviewer 2 Report

Comments and Suggestions for Authors

General

This paper offers some elegant mathematics and interesting calculations for 2D array of periodically spaced identical plates (screens) but its relevance to the practical application of louvred screens for windows is not clear. The only limitation mentioned at the bottom of page 11 in the Discussion section is that in many practical applications the plates (screens) would have finite ‘height’. Incidentally the description finite ‘width’ would be more appropriate. However, no account is taken of the influence of window frame and reveals. This is not only a major shortcoming, but it misses any consideration of the potential contribution of the window frame construction to transmission loss.

A problem with solutions based on periodic arrays such as considered by the author is that they result in highly fluctuating frequency dependent transmission loss [see for example the review in Rubino et al Balancing ventilation and sound insulation in windows by means of metamaterials: A review of the state of the art, Building and Environment, 275, (2025), 112780 https://doi.org/10.1016/j.buildenv.2025.112780 ].

Different approaches offer much more broadband transmission loss [Orduña-Bustamante et al Simplified vented acoustic window with broadband sound transmission loss, Applied Acoustics, 217, (2024), 109865, https://doi.org/10.1016/j.apacoust.2024.109865 ] and significantly higher transmission loss than suggested the author [see for example https://etheses.whiterose.ac.uk/id/eprint/30239/1/Fusaro_Gioia_170244679_rev1.pdf ] .

Given the highly theoretical emphasis of the present paper, a more suitable journal might be Wave Motion. Indeed, several of the author’s references are from Wave Motion.

Details

Line 36 I suggest replacing ‘same’ by ‘identical’

Line 87 I suggest replacing 'latest' by 'latter'

Line 135 delete ‘are found’

Line 235 Although ‘active’ is the opposite of ‘reactive’ it is more usual to refer to an impedance with zero imaginary part as ‘resistive’.

Comments on the Quality of English Language

See 'details' in comments to author

Author Response

Dear Reviewer

I would like to extend my sincere gratitude for your thoughtful comments and for the tip you provided regarding acoustic window applications. To be honest, I had not previously considered this specific application of my research for acoustic windows, but I understand its significance. I believe it should be included in the submitted paper and I appreciate your input. Please, find below detailed replies to your remarks.

This paper offers some elegant mathematics and interesting calculations for 2D array of periodically spaced identical plates (screens) but its relevance to the practical application of louvred screens for windows is not clear. The only limitation mentioned at the bottom of page 11 in the Discussion section is that in many practical applications the plates (screens) would have finite ‘height’. Incidentally the description finite ‘width’ would be more appropriate. However, no account is taken of the influence of window frame and reveals. This is not only a major shortcoming, but it misses any consideration of the potential contribution of the window frame construction to transmission loss.

In my view, for practical use, it is important to correctly choose the acoustic impedance of screens. To understand the physical aspects, it is best to consider the problem analytically. Diffraction problems are often not very simple, so I decided to analyze an ideal infinite structure. This gave some results on how the impedance affects the scattered sound fields.

I agree that finite barriers should be taken into consideration for practical purposes. By "height", I mean the limited height of a barrier (for example, from ground level to barrier top as shown in Figure 10 of the revised paper). You are correct about "width", as in real-world scenarios we often deal with limited barrier sizes. In my opinion, analytical solutions for finite 3D structures can be very challenging, and these structures should be simulated. This seems like the next step in the research process. This issue has been added to Discussion as an example of the perspective development.

A problem with solutions based on periodic arrays such as considered by the author is that they result in highly fluctuating frequency dependent transmission loss [see for example the review in Rubino et al Balancing ventilation and sound insulation in windows by means of metamaterials: A review of the state of the art, Building and Environment, 275, (2025), 112780 https://doi.org/10.1016/j.buildenv.2025.112780 ].

Thank you very much for this reference. I have really missed acoustic window applications, but it is a very perspective issue. As I understand it, highly fluctuating frequency-dependent transmission loss is connected to high-order modes present in the channel. In the submitted paper, I limited the structure period to be no greater than half the wavelength, so high-frequency resonant transmission was outside the scope of the manuscript. A reference to this interesting review has been added to the revised version of the paper.

Different approaches offer much more broadband transmission loss [Orduña-Bustamante et al Simplified vented acoustic window with broadband sound transmission loss, Applied Acoustics, 217, (2024), 109865, https://doi.org/10.1016/j.apacoust.2024.109865 ] and significantly higher transmission loss than suggested the author [see for example https://etheses.whiterose.ac.uk/id/eprint/30239/1/Fusaro_Gioia_170244679_rev1.pdf ] .

Thank you again for the references. They are very useful for my research. I agree that the authors achieved significant transmission loss over a wide frequency range, especially in the mid-frequency band. It seems that there may be more effective methods than the screen with uniform impedance considered in the paper.

Given the highly theoretical emphasis of the present paper, a more suitable journal might be Wave Motion. Indeed, several of the author’s references are from Wave Motion.

I appreciate your advice. Wave Motion is a great journal, and I will definitely consider submitting my next paper to it.

Details

Line 36 I suggest replacing ‘same’ by ‘identical’

Line 87 I suggest replacing 'latest' by 'latter'

Line 135 delete ‘are found’

Line 235 Although ‘active’ is the opposite of ‘reactive’ it is more usual to refer to an impedance with zero imaginary part as ‘resistive’.

All suggestions have been taken into consideration for the revised manuscript.

Reviewer 3 Report

Comments and Suggestions for Authors

The article entitled "Diffraction of sound waves by a periodic array of impedance screens" theoretically approaches the problem of diffraction of sound waves on periodic structures. The author's approach is rigorous and original, anchored in good quality and numerous references. Regarding the theoretical modeling and the possibility of optimizing the structures, there is nothing to object to or improve. In my opinion, this article, otherwise very interesting and well-written, suffers from the lack of experimental validation or at least a subchapter as a comparative study. Objectively, here are some observations that in my opinion can bring added value:
1. The article has no Conclusions;
2. The article is summarized as a very good quality theoretical study without a comparative study to offer readers a relevant image of the contribution;
3. I believe that information is needed regarding the simulation - software support, computational effort in a practical case would bring a plus;
4. Regarding 3, there could be a discussion related to the efficiency of the model (computational effort vs. quality of results) which would bring a plus;
5. It is important to provide sufficient technical details regarding the implementation of the simulation for readers interested in reproducing the results or evaluating them with other approaches.
I believe that a major revision that would give your theoretical study a utility on a standard case would be beneficial and would bring visibility to this contribution.

Author Response

Dear Reviewer,

I am grateful for your evaluation of the manuscript. Due to your comments, the manuscript has been revised. Specifically, I have expanded the "Discussion" section with the potential future development of the study and have added "Conclusions". Please, find below detailed replies to your remarks.

The article entitled "Diffraction of sound waves by a periodic array of impedance screens" theoretically approaches the problem of diffraction of sound waves on periodic structures. The author's approach is rigorous and original, anchored in good quality and numerous references. Regarding the theoretical modeling and the possibility of optimizing the structures, there is nothing to object to or improve. In my opinion, this article, otherwise very interesting and well-written, suffers from the lack of experimental validation or at least a subchapter as a comparative study. Objectively, here are some observations that in my opinion can bring added value:

1. The article has no Conclusions;

According to the author guidelines the section “Conclusions” is not mandatory but can be added to the manuscript if the discussion is unusually long or complex. However, due to the recommendations of the reviewers, I have added some information about perspective problems to the "Discussion" section. This has made the "Conclusions" more relevant, and this section has been included in the revised paper.

2. The article is summarized as a very good quality theoretical study without a comparative study to offer readers a relevant image of the contribution;

Thank you for your remark on this important point. I completely agree that a comparison with other studies would be both useful and interesting. While I had considered this idea while preparing the manuscript, it was not as straightforward as I hoped. In the experimental studies, there is no information available about the acoustical impedance of the screens that were tested, which makes it impossible to accurately compare the presented theory with the open experimental data. Nevertheless, I hope that the results of this paper will contribute to the theoretical understanding of this issue and lead to a more comprehensive comparative study in the future.

3. I believe that information is needed regarding the simulation - software support, computational effort in a practical case would bring a plus;

Actually, simulations were not used in the preparation of the manuscript. Instead, only the linear system (13) was solved numerically. This was a routine calculation that could be performed using different software programs. In this study, I used Mathcad without any special adjustments. Therefore, I do not believe that my work contributed significantly to the computational aspects of the research.

4. Regarding 3, there could be a discussion related to the efficiency of the model (computational effort vs. quality of results) which would bring a plus;

The physical model has been simplified into a system of linear equations (13), which can be solved using various methods. In this case, simulation was not necessary, and I believe this is the most significant achievement in the analytical section of the paper. The calculation time was relatively short, with the longest period (approximately a few seconds) needed to obtain 1,000 solutions shown in Figure 2. This system (13) can be solved with any desired level of accuracy without requiring specialized simulation or computation techniques.

5. It is important to provide sufficient technical details regarding the implementation of the simulation for readers interested in reproducing the results or evaluating them with other approaches.

Thank you for your feedback on this point. As I mentioned, the calculation did not require significant computational resources, and I have added a comment about this in the beginning of Section 3. I agree that the results could be reproduced using different calculation methods. However, I would prefer not to discuss the calculation aspect further in the manuscript, as there are no new findings in this area.

I believe that a major revision that would give your theoretical study a utility on a standard case would be beneficial and would bring visibility to this contribution.

Round 2

Reviewer 2 Report

Comments and Suggestions for Authors

Thank you for the revisions which make clearer the limited practical contribution of the work in the context of windows and indeed that there are more successful ventilated window designs.

Given the limitations of application to windows, I wonder why the author has not considered the potential relevance of the work to the design of louvred outdoor noise barriers. However it is not essential to mention the possibility at this stage.

Reviewer 3 Report

Comments and Suggestions for Authors

The author of the article in this new version corrects some aspects that were the subject of constructive comments in the first version. The article in this new version, in my opinion, meets the criteria for publication.