Resonant Metasurfaces with a Tangential Impedance

Round 1

Reviewer 1 Report

The authors performed theoretical derivation of a new concept, “tangential impedance”, for resonant metasurfaces. This is an interesting work as tangential acoustic response could be important for this kind of metasurfaces. In my opinion, the main shortage of this work is its lack of experimental validation. I think this should be addressed by at least showing that the present model captures key physics compared against experimental results. If this could be provided, I will be very happy to suggest its acceptance by the journal.

Author Response

Dear Reviewer,

Thank you for the positive assessment of the manuscript. I agree that it would be useful and interesting to carry out the experiment validation. But I have to state that experiments are not in the scope of the presented research, which focuses on the theoretical aspects of the problem. On the other hand, I hope that the results will attract attention of researchers dealing with the experiments and new data will be captured (and then published) for structures similar to considered in the manuscript.

Reviewer 2 Report

This is more theoretical work in nature with more focus on formulations. There are no numerical results included for comparison purposes. The article can be accepted after revision. (1) Can the author validate the theoretical findings with a numerical code or some other approach? (2) The author should proofread the paper for English grammar issues on page 1, line 41 "do change" is "to change" and on page 9, line 287, "is have", I think "is" is extra. Likewise, across manuscripts, authors should check for such grammar issues.

Author Response

Dear Reviewer,

Thank you very much for your opinion and suggestions.

(1) Can the author validate the theoretical findings with a numerical code or some other approach?

The main goal is to demonstrate the possibility of unusual reflection properties of metasurfaces. The problem statement has been idealized (an infinite regular metasurface and a plane harmonic incident wave, point dipole resonators) in order to obtain an exact analytical solution. The solution shows obviously the physics and allows to propose a special boundary condition for the metasurfases of a similar type.  However, I am sure that numerical simulations will be useful for next steps of the research when we (or somebody else) study the more realistic structures taking into account practical limitations.

(2) The author should proofread the paper for English grammar issues on page 1, line 41 "do change" is "to change" and on page 9, line 287, "is have", I think "is" is extra. Likewise, across manuscripts, authors should check for such grammar issues.

I have checked the text. Some typos and mistakes have been found and corrected. Thanks!

Reviewer 3 Report

see the attachment

Comments for author File: Comments.pdf

Author Response

Dear Reviewer,

I am very grateful for the careful reading of the article and given remarks. Please, find my comments below

1，there is no space in front of “where”, such as line 83, 93, 95....

All spaces in front of “where” was deleted.

2，Please double check the Eq. (3), there is a “i” in the first term , if it is correct.

Yes, “i” is correct because I use the definition for the vertical component α_m (now η_m) of the wave vector having real values for non-uniform waves. The goal is to have real and positive values in the series for X (equations 7, 13, 25, 30).

3, “component” on line 83 should be “components”

Corrected. Thank you!

4, The work proposed is based on the idea assumption, are there any numerical results to show phenomenon of the proposed work.

The main idea of the paper is to show an opportunity of unusual acoustical reaction of a boundary. For this reason, the problem is formulated to obtain an exact and clear solution. In my opinion some simulations will be needed when more complex systems are being studied.

5, The work show a good theatrical background of the meta material in the introduction part, the work would be better presented if the related numerical works can be mentioned.

Thank you for the references. The works have been cited.

Reviewer 4 Report

The author discussed the tangential impedance of the structure with diploe resonators which is interesting. But the following questions should be modified before the article is accepted.

1. Usually, the subwavelength means the thickness of the metasurface which is not the distance of the neighbored resonators as stated in the manuscript. Please check this conception. Maybe metamaterial is more suitable.

2. How to change the values of Z_⊥or Z_∥ ?

3. The author just give some different boundary conditions to obtain Figs. 6-7. How to realize?  This question is a little similar to the above one.

Author Response

Dear Reviewer,

I would like to thank you for your attention for the paper and useful tips. My answers to your questions are below. The changes are marked in the text.

1. Usually, the subwavelength means the thickness of the metasurface which is not the distance of the neighbored resonators as stated in the manuscript. Please check this conception. Maybe metamaterial is more suitable.

In the studies problem the resonators are small in comparison with a wavelength, moreover the membrane resonators has the zero thickness. So, we consider very thin layer, which is actually the boundary for the incident wave.

As for the distance between the resonators, if it is smaller than a half wavelength, the result does not depend on the distance. The membrane resonators have finite size; therefore, we can construct the structure without gaps between the resonators. We should just suppose L=2a. To underline this point I have added the comment: “Note that the membranes in both metasurfaces (Figures 4 and 5) can be placed without gaps between each other. We should only suppose L=2a to obtain the formulas for this case.”

Conceptually we deal with the boundary formed by the resonators (meta-atoms) and I guess that the presented structures are better called metasurfaces.

2. How to change the values of Z⊥or Z∥ ?

Both impedances can by adjusted by varying physical parameters of the resonators. For example, the real part of the impedances depends on the friction in the resonator and the image part depends on the stiffness in the dipole resonator and the tension of the membrane resonators.

To obtain the real value impedance we should adjust the eigenfrequency of the resonator so the satisfies the condition Im(Z+Zr)=0. We can use materials with different dissipation (image part of the Young’s modulus) for membranes to change the real part of impedance.

3. The author just give some different boundary conditions to obtain Figs. 6-7. How to realize? This question is a little similar to the above one

To make clearer previous and this points I have added this comment:

The impedances of the resonant metasurfaces are real if the resonant frequency found from the equations “Im(Z⊥+Z)”=0 or "Im(Z||+Z)=0” coincides with the frequency of the incident wave. The value of the real part can be adjusted by varying dissipation properties of the resonators. For example, the membranes could be manufactured from elastic materials with different loss coefficients, which are described by the image part of the modulus of elasticity.

Reviewer 5 Report

The manuscript elaborates around the scattering produced by an acoustic metasurface composed of resonant elements (monopoles and dipoles) over a rigid surface. Then, the manuscript express the reflection coefficient in terms of a impedance normal to the interface (for the monopoles inclusions), but, interestly, a tangential impedance for the dipole resonators. The manuscript shows potential in terms of explaining the interaction of resonant particles for acoustic waves. However, there are relevant issues that makes this manuscript not suitable for publication for Acoustics in its current form.

The main concern related to this manuscript is its lack of numerical or experimental validation. While the manuscript performs a systematic analysis of the interaction between an incident wave in a interface made of periodic resonant structures, the only information provided comes from the analytic expressions themself. Therefore, it is required at least a numerical simulation of the proposed structures that proves the normal excitation of the monopoles resonators and the tangential excitation of the dipole ones.

In addition, the author provides two definitions of the tangential impedance, one based on the reflection coefficient of Eq. (17), and Eq. (32) based on the second derivative of the pressure wave. Without simulation or experimental results, it becomes impossible to determine which is the proper definition.

In page 8, line 269, it is proposed an alternative definition of the normal impedance where it is implied that it is produced by the normal component of the pressure. This is done to present the alternative definition of the tangential impendance. Before that, it was stated that the normal impedance was named like that since it depends on the normal component of the velocity (like also the tangential impedance). Please clarify the role of the normal impedance.

There are minor misprints in page 1, line 41 (... are used to change...); and in page 2, line 65 (...properties of the metasurfaces...).

In page 9, line 307, the symbol of absoprtion alpha is referred as the reflection coefficient (referred as V earlier in the manuscript). Please fix the mistake.

Author Response

Dear Reviewer,

Many thanks for your efforts to review the paper and remarks concerning its content.

#1 The main concern related to this manuscript is its lack of numerical or experimental validation. While the manuscript performs a systematic analysis of the interaction between an incident wave in a interface made of periodic resonant structures, the only information provided comes from the analytic expressions themself. Therefore, it is required at least a numerical simulation of the proposed structures that proves the normal excitation of the monopoles resonators and the tangential excitation of the dipole ones.

Numerical simulations have not been included in the current stage of the study. The first step is to propose a concept of metasurfaces, which react tangentially to the acoustic impact. So, the problem was formulated in a such way to find the solution analytically. The found answer are the direct and exact consequences from the wave equation and the equations of motion. Moreover, I believe that these solutions can be used for verification of numeric models and simulations at the next stages of the research.

#2 In addition, the author provides two definitions of the tangential impedance, one based on the reflection coefficient of Eq. (17), and Eq. (32) based on the second derivative of the pressure wave. Without simulation or experimental results, it becomes impossible to determine which is the proper definition.

Actually, these equations are just different definitions of the tangential impedance. Eq. (17) and (31) are found for the periodic structure of the resonators and take into account the design of the structure (the type of the resonators, its sizes and the period of the array). Eq.(32) is proposed for the general case of the tangentially reacting surfaces. If the surface behaves like the array of the dipole resonators we can use Eq.(32) without taking into account the microstructure of the metasurface.

It will be very convenient when we come to the next stages of the study of the tangential impedances. For example, the problem of sound propagation in ducts with the tangential impedance walls or the problem of the sound reflection of a point source from the dipole metasurface can be solved by using the simple boundary condition given by the equations like Eq. (32) instead of considering the complex inner structure of the metasurface.

#3 In page 8, line 269, it is proposed an alternative definition of the normal impedance where it is implied that it is produced by the normal component of the pressure. This is done to present the alternative definition of the tangential impendance. Before that, it was stated that the normal impedance was named like that since it depends on the normal component of the velocity (like also the tangential impedance). Please clarify the role of the normal impedance.

You are right the term “normal impedance” is used only to differentiate it from the new impedance. So, the normal impedance is the ordinary impedance of the surfaces which is usually used in acoustics. It is not a new concept but very traditional value. The mark “normal” means that the sound pressure produces the force which acts normally to the surface, but the sound pressure is a scalar value and it has not the normal or tangential component.

The normal velocity of the surface is a reaction of the surface to acoustic impact. This value is used for description of the surface reaction in both definition of the impedances (normal and tangential). The words “normal” and “tangential” show the type of impact on the boundary, whereas the reaction to this impact is the same in both cases.

#4 There are minor misprints in page 1, line 41 (... are used to change...); and in page 2, line 65 (...properties of the metasurfaces...).

They are corrected. Thank you!

#5 In page 9, line 307, the symbol of absoprtion alpha is referred as the reflection coefficient (referred as V earlier in the manuscript). Please fix the mistake.

The letter alpha is usually used for the absorption coefficient. Therefore, I changed the first definition for the symbol alpha to the symbol eta.

Round 2

Reviewer 1 Report

I am happy to suggest its acceptance for publication in its current form.

Reviewer 4 Report

The questions are modified by the author and the paper can be accepted now.

Reviewer 5 Report

The reviewer is pleased with the comments and corrections presented by the author.