Reply to Minin, I.V.; Minin, O.V. Comment on “Mahariq et al. Investigation and Analysis of Acoustojets by Spectral Element Method. Mathematics 2022, 10, 3145”

Abstract

This reply provides one-to-one responses to comments regarding the article entitled: “Investigation and Analysis of Acoustojets by Spectral Element Method” that has been published in Mathematics by MDPI. As a corresponding author, I am happy to receive constructive criticism about the aforementioned article, in particular, suggestions and/or comments that, when addressed, may further help the reader in understanding the published material.

1. Introduction

This reply provides responses to discussion related to the abovementioned article [1].

In regard to the two introductory paragraphs, I would like stress the fact that this work has been published in a mathematics journal, i.e., the Spectral Element Method (SEM), its mathematical formulation and the numerical solution are the main thrust and concern of our article. Commenters have requested more details regarding the m factor, which was derived under assumptions that are beyond the scope and remit of this paper. This factor depends on parameters that are more relevant to mechanics, and so, we believe details about this would grossly detract from the main message of the paper. We did use the word “index”, but it does not necessarily hold the same meaning as “refractive index” used in optics, which is limited to speed ratio. We define the m factor, in the published article [1], as a constant number, or a normalization factor, which depends on relative pressure, speed and acoustic impedance. Therefore, “index” is used to express a uniquely defining parameter, for a given material, that accommodates its properties and characteristics; thus, is not necessarily merely an expression of speed ratio.

Regarding the third paragraph in the comments [2], although we did not use the term “polarization” in our article, it is noteworthy that the problem of scattering in a homogeneous object placed inside a homogeneous medium can be solved based on the Helmholtz equation in the harmonic case, as in [3]. Specifically, we solved the Helmholtz equation in its scalar form. In other words, the considered problem is a two-dimensional one, and, as mentioned by the reviewer, there are some cases where isotropic characteristics can be considered. Hence, since our contribution, in the first place, is the mathematical formulation of SEM, acoustojet was aptly considered an example for verification.

Regarding the sixth paragraph, recall the following statement from the original article [1]: “In analogy to the nanojet effect, subwavelength beam localization is also possible for acoustic and ultrasound fields, which is termed as “acoustic jets” or “acoustojets ”[4–10]”. The inclusion of this statement is based on the references provided, and emphasizes the analogy between photonic jets and acoustic jets and their characteristics, in general. In particular, with regard to the use of the term “acoustic jets”, specifically, this term is used in “[6,7]”; in fact, this term is used in the titles of those references. In addition, both the terms “acoustic jets” and “acoustojets” were used in “[9]”. These articles were referenced in our original article. Herewith, another reference is provided [4] in which the term “Acoustojet” is used.

Regarding the lines 64–69, the mentioned comments concluded from “[17,18]” were included in the literature review part, although they refer to synthetic jets. It is noteworthy that the literature review in a typical article may cover the experimental, theoretical, and synthetic aspects of the topic of interest.

2. Conclusions

In conclusion, we do welcome any constructive comments that help the readers to better understand the published topic; however, some of the comments provided by the commenters may serve to detract the reader’s attention from the main contribution detailed in the article [1].

Finally, we would like to thank the reviewers, commenters and the editor for their valuable and valued comments and suggestions regarding our manuscript. We very much appreciate your time and effort.