Local Splitting into Incoming and Outgoing Waves and the Integral Representation of Regular Scalar Waves

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

Report on: On the local splitting into incoming and outgoing waves and the

integral representation of regular scalar waves

 

Authors introduced and setup a novel problem on the splitting into an incoming field and

an outgoing field and established that for a regular field such splitting is in general not possible near to given surface. They considered a simple case of a two-dimensional problem with a circle as the curve Γ. They also addressed the problem of the possibility of representing a regular field as an integral over curve Γ that would be valid outside Ω. They also obtained characterization of the fields. The mathematical results/derivations are interesting. The limitation of the present study is that such a representation is not possible in general.  Authors are also encouraged to present some possible applications.    

Suggestion about this manuscript is to expand its introduction as well as references, if possible. Rest of the derivations are okay and well written. However, the problem considered is of specific nature. If addition of more citations is not possible, then authors can work only on the introduction part to add some more details about the problem formulation, novelty, applications, etc. 

Author Response

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Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

Review Report

 

Title: On the local splitting into incoming and outgoing waves and the integral representation of regular scalar waves

 

Comments to the author: I have reviewed the paper. I have the following major concerns:

 

Major comments:

 

• The introduction must be separated with the formulation of the problem. In the introduction, the author should clearly explain the applications, the purpose of this study, novelty and the application of the proposed problem.
• The author claimed in the manuscript that “splitting into incoming and outgoing fields is not possible in general” but it didn’t describe it precisely.
• A theorem stating the sufficient and necessary conditions for splitting, but no examples have been provided that exist beyond the plane wave.
• The two dimension Bassel and Hankel series argument mixes point wise convergence and weak convergence, but they didn’t clearly define the exact function space framework.
• The author didn’t clearly explain in which failure occurs i.e., the jump among the L2 criteria and Schwartz distributions.
• In Lemma 1 the Jordan integral equations represents the operator K is not invertible because 0 lies in the essential spectrum. This only reveals that ill-posedness, but it isn’t strictly impossible, we can still obtain the solutions for u that satisfies the Picard condition. This conflicts the ill-conditioning and non-existence. A clear explanation is required for the difference.
• The author should clearly explain what happens at resonance when k^2 is an eigenvalue.
• A plot of eigenvalue and mechanism of convergence is necessary for enhance the credibility of the manuscript.
• Many statements are given in analytical forms, but numerical outcomes have been ignored.
• Conclusion is very short, the author should explain the limitations and possible future work about the proposed work.

 

Author Response

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Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

See the attached file.

Comments for author File: Comments.pdf

Author Response

Please see the attachment

Author Response File: Author Response.pdf

Round 2

Reviewer 2 Report

Comments and Suggestions for Authors

it can be accepted now for possible publication. 

Reviewer 3 Report

Comments and Suggestions for Authors

Referee report: On the local splitting into incoming and outgoing waves and the integral representation of regular scalar

waves (revised)

 

By Didier Felbacq and Emmanuel Rousseau

    

The revised paper by Didier Felbacq and Emmanuel Rousseau is in its present form suitable for publication in your journal. The authors have replied to my questions and objections in quite a satisfactory way. The alterations and additions in the manuscript have provided answers with respect to the fundamental questions, with which were not dealt in the previous version of the paper. The theory, developed in the manuscript, is put into a broader perspective.

Therefore, I will recommend the publication of this thoroughly written paper, which will provide a valuable contribution to this interesting problem.