Complex Dispersion Relation Recovery from 2D Periodic Resonant Systems of Finite Size
Round 1
Reviewer 1 Report
The paper is easy to read and follow. I recommend its publication. There are a few minor text corrections that should be addressed before the paper is published. Some of the obvious errors are highlighted in this review but the authors are encouraged to carefully proof read their manuscript.
Abstract, line 3, it is unusual to see references in the abstract. Further, the reference number is [14] but perhaps should be [1]
Abstract, line 10, space needed between: even if
Page 1, line 31, why do the references jump from 10 to 14? The references should increase in consecutive order.
Page 2, lines 48-53. This sentence is too long, rewrite for clarity.
Page 2, line 53: spatial Laplace transform (LT)
Page 3, lines 71/72, are the authors referring to the correct figure number here? I think they are may be referring to Fig. 2.
Page 4, line 9, a temporal and a spatial FT.
Page 5, line 116, rotated by 45
Page 6, line 131, Bragg
Page 7, line 151, for a periodic system of infinite extent.
Page 7, line 155, rapidly
Page 7, line 168, symmetric with respect to
Page 8, line 182, plane
The literature review could include further relevant references on finite sized sonic crystal arrays (with and without locally resonant scatterers), for example, see Montiel et al. Wave Motion 70 (2017) and references therein.
Author Response
The authors would like to thank the constructive comments and suggestions from the referee. In the attached letter both the answers to his/her questions and the changes performed in the manuscript are shown. In cursive bold characters we show the comments of the referee, in normal characters we show our comments and responses. In blue cursive characters we show the text we have added to the manuscript.
The paper is easy to read and follow. I recommend its publication. There are a few minor text corrections that should be addressed before the paper is published. Some of the obvious errors are highlighted in this review but the authors are encouraged to carefully proof read their manuscript.
The manuscript has been intensively revised to correct text errors suggested by the referee. Changes are highlighted in red in the revised manuscript.
The literature review could include further relevant references on finite sized sonic crystal arrays (with and without locally resonant scatterers), for example, see Montiel et al. Wave Motion 70 (2017) and references therein.
We have included further references on finite size periodic systems at the end of the introduction. The text added to the manuscript reads as follows:
"Moreover, the SLaTCoW method is particularly suitable for the experimental characterization of real periodic systems of finite-size (resonant or non- resonant), which, while exhibiting similar properties as systems of infinite extent such as band gaps [16-18] or other dispersive effects [19, 20], can not be experimentally characterized using methods considering infinite structures, such as EPWE, MST or solving an eigenvalue problem in FE."
Author Response File: 
Author Response.pdf
Reviewer 2 Report
The authors present a new method to calculate complex band gap (dispersion relation), the effectiveness of the presented method is approved by comparing plane wave expansion. As a new contribution to numerical methods for metamaterials, it could be published. The authors are suggested to calculate more example and compare with finite element method.
Author Response
The authors would like to thank the constructive comments and suggestions from the referee. In the attached letter both the answers to his/her questions and the changes performed in the manuscript are shown. In cursive bold characters we show the comments of the referee, in normal characters we show our comments and responses. In blue cursive characters we show the text we have added to the manuscript.
The authors present a new method to calculate complex band gap (dispersion relation), the effectiveness of the presented method is approved by comparing plane wave expansion. As a new contribution to numerical methods for metamaterials, it could be published. The authors are suggested to calculate more example and compare with finite element method.
While the main purpose of the present work is to validate the proposed method for 2D metamaterials, i.e., 2D periodic systems with resonant scatterers, we have included in the manuscript an example of a 2D finite-size periodic system made of non-resonant scatterers. In this regard, the results of this example are compared to the well known EPWE. We would like to point out that the method is completely valid for other type of non-resonant periodic systems with different parameters, such as different lattice constants or other type of lattices, triangular, etc. Moreover, as shown in Ref. [11] of the main text, where the SLatCoW method was first presented, different systems are presented and results using the SLaTCoW method are fully validated.
Regarding the comparison of the results with finite element method, we would like to note that the complex dispersion relation results presented for the 2D periodic resonant system in Section 4.2 are indeed compared to FEM simulations. These results are labeled as "Theory" in Fig. 4 of the manuscript and the method used is described in the text:
"The real part of the wavenumber is shown in the central panel and results obtained using SLaTCoW are compared to those obtained solving an eigenvalue problem using FEM over an infinite periodic structure with the same geometric and physical characteristics."
Author Response File: Author Response.pdf
Reviewer 3 Report
The authors show a methodology to calculate the dispersion relation of periodic resonant and non-resonant systems. They validate the results show an agreement with the theoretical results.
I recommend the manuscript for publication after minor changes.
- The study of the limitations of the method must be increase making a study of the quantity of losses are reproducible and the frequency range with respect the lattice parameter.
- The author can consider if is this method valid for other kind of waves as electromagnetic waves?
Minor changes
The manuscript English must be revised (i.e. Braag instead Bragg)
The references must be revised.
Author Response
The authors would like to thank the constructive comments and suggestions from the referee. In the attached letter both the answers to his/her questions and the changes performed in the manuscript are shown. In cursive bold characters we show the comments of the referee, in normal characters we show our comments and responses. In blue cursive characters we show the text we have added to the manuscript.
The authors show a methodology to calculate the dispersion relation of periodic resonant and non-resonant systems. They validate the results show an agreement with the theoretical results.
I recommend the manuscript for publication after minor changes.
The study of the limitations of the method must be increase making a study of the quantity of losses are reproducible and the frequency range with respect the lattice parameter.
The main purpose of the present work is to validate the proposed method for 2D metamaterials, i.e., 2D periodic systems with resonant scatterers. In the current work there are two systems compared with two different theoretical models showing the validity of the method. Referee is asking for a key problem of the current work: a parametric study of the method in terms of the amount of losses in the system introduced by reducing the lattice parameter while fixing the radius of the inclussions. The main problem face to this parametric study is that there are not analytical/numerical methods working with losses in the regime of closely packed sonic-crystals or metamaterials around the frequencies of the Bragg interferences. The available models are based on homogenized theories that are far away from the Bragg frequencies. Therefore, the only theoretical comparison we can perform could be based on the numerical solution of the full problem solving the Navier-Stokes equation or the implementation of the full Multiple Scattering Theory solving both the acoustic and thermal parts of the problem. In both cases we are not able to perform now the comparisons. Other possibility consists of the experimental validation. Concerning the experimental validation, as shown in Ref. [11] of the main text, where the SLatCoW method was first presented, different systems are presented and results using the SLaTCoW method are fully validated experimentally for single incidence directions. There is no reason to extend the conclussions of this paper to higher dimensions. We would like to point out that the method is completely valid for other type of non-resonant periodic systems with different parameters, such as different lattice constants or other type of lattices, triangular, etc.
The author can consider if is this method valid for other kind of waves as electromagnetic waves?
Indeed, the acoustic wave equation in 2D is equivalent to TE and TM polarizations for electromagnetic waves. Hence, this method should be perfectly valid for these type of waves. We have added the following text in the manuscript to highlight this point:
As shown in Refs. [11, 12] this method is applicable to systems of different kind, at different frequency ranges and should also be applicable for other type of waves, such as TE and TM modes for electromagnetic waves, where there is a close similarity with the 2D acoustic wave equation.
The manuscript English must be revised (i.e. Braag instead Bragg)
The manuscript has been intensively revised to correct text errors and improve spelling and correctness. Changes are highlighted in red in the revised manuscript.
{The references must be revised.
We have fully revised the references, including their format, as well as their suitability and correct ordering within the text. Moreover, following the suggestions from other reviewer, we have added further references in the manuscript, see below the text added to the manuscript:
"Moreover, the SLaTCoW method is particularly suitable for the experimental characterization of real periodic systems of finite-size (resonant or non- resonant), which, while exhibiting similar properties as systems of infinite extent such as band gaps [16-18] or other dispersive effects [19, 20], can not be experimentally characterized using methods considering infinite structures, such as EPWE, MST or solving an eigenvalue problem in FE."
Author Response File: Author Response.pdf
