Efficient Modelling of Acoustic Metamaterials for the Performance Enhancement of an Automotive Silencer

Round 1

Reviewer 1 Report

This paper presents a study on the usefulness of acoustic metamaterials to reduce noise annoyance from an automative silencer. The authors present a model for a metamaterial, focussing their work on different analysis (analytical model, experimental set-up, numerical simulation), to enhance the attenuation of the sound waves on a given frequency range. The low cost in the design phase of such metamaterial-made silencers in the industry is the motivation of this work. They conclude that their model shows that acoustic metamateriales can improve sound attenuation. The study presented here is interesting and may be of interest for a large community (engineers, researchers). The paper is structured correctly. 

Before publication of this manuscript in Acoustics, I suggest to improve some aspects of the description of the work. These points are the following. 

1) Some references should be added in the first two paragraphs of Section 1. 

2) A reference should be added when COMSOL is mentioned for the first time. 

3) All the symbols must be defined when used for the first time (W_i, r, f_i, u_i, etc.), especially in all the equations and in Fig. 4. 

4) Eq. (1): dB can be removed from the equation since it is explained in the following sentence. 

5) Paragraph Line 100: explain, briefly, what the method used is, if not with equations, al least how COMSOL is used for this purpose (a schematic diagram may be useful). 

6) Paragraph Line 103: explain or rewrite this paragraph. 

7) Eq. (2): T_2 should be T_22. 

8) Paragraph after Eq. (2): explain "By changing the load conditions". 

9) Index for outlet is 0 or r (Eqs. (1) and (3))? 

10) Line 118: a reference should be added for standards. 

11) Sections 2 may include two subsections for Numerical method and Experimental set-up, respectively. 

12) Fig. 3 should be cited in the text (Paragraph Line 187?). 

13) Paragraph Line 194: what would be the consequences on sound attenuation if materials suitable to a real automobile exhaust system were used? A comment should be added. 

14) Line 232: d or f? 

15) Paragraph Line 240: a reference to a previous section and/or figure may be needed. 

16) Fig. 7 must be cited in the text (Paragraph Line 247?). 

17) The huge differences observed in Fig. 6 deserve more than only one sentence Line 257. 

18) Line 290: "excellent agreement" is not appropriate. 

19) Fig. 13. Name each cylinder type in the figure caption. 

20) Paragraph Line 324: Explain and justify the differences observed between experimental and numerical data. The measurement limit is not convincing in my opinion. 

21) Line 331: "good agreement" is not appropriate (not that clear in Fig. 16). 

22) Line 332: why "potentially"? 

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Reviewer 2 Report

This is an interesting paper on the application of the emerging metamaterial technology to automotive silencer design.

However, some points need to be clarified before full acceptance:

1. In lines 142-144 it is written “This geometry gives low transmission loss at frequencies over 500 Hz. This makes it easier to accurately measure the transmission loss values experimentally as the frequencies of interest lie within the operating range of the test rig”. This seems strange, because it easier to accurately measure the transmission loss values when they are high, not when they are low (?). Please clarify.
2. It is not specified what D(omega) is in eq. (4). Of course it can be deduced from the previous exposition, however, for the sake of clarity it is better to explicitly define D(omega) adding a further equation. Moreover, it could be useful for the readers to add an annex showing the explicit form of the wall impedance Z.
3. In lines 259-273, the authors summarize a fitting procedure to adjust the internal dimensions of the metamaterial, within manufacturing tolerances, until a match was achieved with experimental results at grazing incidence. However, the wall impedance Z was analytically derived at normal incidence. So, the reader may wonder whether the incidence angle should also be considered. Some further clarification is needed here.
4. In line 300, it is better to not speak of “the excellent match between the experimental and numerical data”, as in the line above it is written “The numerical results assume a lossless system with acoustically hard wall, which cannot be replicated precisely in the physical set up”. A “good/acceptable match, considering the expected differences” is suggested.

Minor issues:

1. Line 110: “transfer mix” should read “transfer matrix”.
2. Line 209: “helmholtz resonators” should read “Helmholtz resonators”.
3. Line 211: “adiabatic dilatation” should read “adiabatic dilatation”.
4. Figure 4 appear “squeezed” in the horizontal direction, so that the circular chambers become ovoidal; please correct.
5. Lines 261-262: “The DENORMS labyrinthine cavity block was set up shown in figure 8 and tested…” should read “The DENORMS labyrinthine cavity block was set up as shown in figures 7 and 8 and tested…”.

In my opinion the manuscript could be ready for publication after a minor revision.

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Reviewer 3 Report

The manuscript addresses the study of the effect of acoustic metamaterials used in automotive silencers by means of a combination of analytical and finite element models, including an experimental validation as well.

In general, the manuscript is well structured, following a logical order and presentation of the work and the results. The provided references are up-to-date and relevant, the quality of the figures is good, and the methodology is adequately described.

Nevertheless, there are some important aspects to be addressed prior to the acceptance and publication of this paper:

1. Lines 21-23: “Unattenuated engine noise can be over 100 dB and this noise level is detrimental to the health of both humans and wildlife”. This statement is widely known in acoustics, audiology, and other related fields. Nevertheless, it could be interesting to include some references to related studies that show how a high SPL affects hearing and health.
2. Lines 74-75: The authors state that “the experimental results will be validated numerically using COMSOL Multiphysics software”. The process should be described the other way around, as it is the experiment the one that validates numerical models.
3. It can be interesting to include, at the end of the introduction, a brief paragraph describing the structure of the paper.
4. (1): W_i and W_r should be defined in the text below the equation, as it is the first time that these parameters appear in the manuscript.
5. Lines 104-106: Why is the two-load method the most suitable method test rig in this case?
6. Line 117: It can be interesting to include a small text (one or two sentences) with a general description of the calibration method. Just writing “switching method” can give too little information for those readers that are not familiar with the topic.
7. Lines 123-124: The authors should justify that the reason for which the cut-off frequency must be considered is because, above this frequency, transverse modes are produced in the acoustic field inside the duct and the propagation of plane waves cannot be ensured.
8. Lines 142-143: “This geometry gives low transmission loss at frequencies over 500 Hz”. The authors should justify this statement according to the physical phenomena induced by the metamaterial having this specific geometry or, alternatively, delete this sentence and provide a more detailed explanation in the results’ section.
9. Figure 4: Please, indicate in the caption the meaning of the different variables appearing in the sketch.
10. (5): The variable r should be defined in the text below the equation, as it is the first time that these parameters appear in the manuscript.
11. Line 219: Are the “dimensions R of the chambers” referred to the radius of the spheres composing the chambers? If so, is should be specified that it is the radius and not the diameter.
12. Section 5: Could the authors provide more information about the numerical model (i.e., resulting number of elements in the mesh, maximum and minimum size of the mesh elements, etc.)? In Figure 9 the mesh looks quite coarse. Was this size enough to obtain accurate results at high frequencies?
13. Line 290: The authors state that the agreement between numerical and experimental results are excellent. I agree with them that the agreement is excellent up to approximately 1140 Hz. However, from this frequency on, some of the peaks of the TL match, but so do not many others. Additionally, the magnitude of the values of the TL obtained by numerical and experimental methods significantly differs. Consequently, I disagree with the statement.

A percentage of error of the numerical results compared to experimental results should be provided in order to confirm the validity of these results.

As a final comment to this point, did the authors include in the simulations the viscothermal losses in the duct and in the slits produced in the metamaterial? Could a finer mesh help to improve the result?

14. In general, the authors describe the results only from the point of view f the agreement numerical-experimental. A deeper analysis of the physical phenomena occurring in the system is missing.
15. There are some small typos in the text, such as “auto motive” (line 56, instead of “automotive”), “transfer mix” (line 110, instead of “transfer matrix”) or “helmholtz" (line 209, instead of “Helmholtz”), among others. Please, check the text to correct these typos.

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Round 2

Reviewer 3 Report

First of all, I want to thank the authors for the work done in the second version of the manuscript. The quality of the presentation and the description of the methods and results has significantly increased.

All my doubts and concers have been clarified, and I have no further comments about the manuscript.