An Unorthodox Arrangement of Boussinesq-Type Wave Equations for Accurate and Robust Numerical Treatment

Response to Reviewer#1 Comments

1. Summary

Thank you for sparing your time for reviewing and making positive comments to improve the manuscript. Please find the detailed responses below and the corresponding revisions/corrections colored in red in the revised manuscript.

Comments and Suggestions to Author

Reviewer: A significant ambiguity in the article is the introduction of surface velocity u₀ at the horizon z=0 in the model of waves with finite amplitude (line 87). After all, at the moment of passage of the trough there are no water at the zero level, which means the speed is not determined. If this is a model for waves of infinitesimal amplitude, then its practical value is questionable.

Author: Use of various different velocities; namely, velocity at the still water level u₀, bottom velocity u_b, mean or depth-averaged velocity , velocity at an arbitrary depth u_α are all common in Boussinesq-type wave modelling. Since virtually all the wave models are developed based on a perturbation process about the still water level z=0, use of u₀ does not introduce any problem concerning its practical applicability.

2. Questions for General Evaluation

Reviewer’s Evaluation

Response and Revisions

Does the introduction provide sufficient background and include all relevant references?

Yes

Are all the cited references relevant to the research?

Yes

Is the research design appropriate?

Can be improved

Improved according to comments

Are the methods adequately described?

Must be improved

Improved according to comments

Are the results clearly presented?

Can be improved

Improved according to comments

Are the conclusions supported by the results?

Can be improved

Improved according to comments

3. Point-by-point response to Comments and Suggestions for Authors

Comment 1: Line 171 – we need to decipher what the index ff is.

Response 1: Thank you for pointing this out. This was a glitch caused by Latex. It was meant to be α hence the velocity u_α which is now put in correct form. Correct subscript is now shown in red color in Line 171.

Comment 2: In Fig. 2 and 4, it is highly advisable to add a drawing with bottom profiles, otherwise readers will have to draw it themselves for a complete understanding.

Response 2: The bottom profiles corresponding to simulations shown in Figs. 2 and 4 are drawn as Fig.3 and Fig.6. Due to addition of new figures all the subsequent figures are renumbered.

Comment 3: Line 378 – it would be necessary to indicate which phase shifts were added manually, otherwise the scale of the disaster is not clear. In Fig. 4, no phase shifts are visible to the eye. It is advisable to show the original data where these phase shifts are visible.

Response 3: The phase shifts introduced are stated quantitatively and the complete paragraph is re-written with more explanations by referring to Hammack’s original comments concerning this problem, as he also had the same problem. On the other hand, drawing both the phase shifted and original computations together becomes quite confusing to the eye; therefore, Fig. 4 (which is Fig. 7 now) is left as before. All these changes are colored red in the text.

Comment 4: Formula 32 should be presented with a picture; it is unlikely that any reader can imagine it without a picture.

Response 4: The bottom profile corresponding to equation (32) was first drawn and placed in the text; however, afterwards it was realized that with a total of 4 more figures the manuscript was overcrowded with figures. Therefore, this newly drawn figure is removed; instead, the reader is directed to the original reference (Beji and Battjes, 1994) where a figure of bottom profile can be found.

Comment 5: Figures 6 and 8 – axes with labels and dimensions are needed.

Response 5: Both Figs. 6 and 8 (now Figs. 9 and 11) are drawn for visual purposes only and both are composed of two different perspective figures (waves and bottom topography) with entirely different scales. Therefore, it is not possible to draw them in true axes (especially in the vertical). Also, since there are two separate figures in one the horizontal axes must be drawn both for waves and bottom topography. Reviewer#5 made a similar comment and asked for color scales for these pictures. Due to reasons indicated above this is not possible either. For clarifying this point the following sentence is added to the captions of these figures: “Axes omitted due to different vertical scales for bathymetry and wave field.”

Comment 6: Figure 8 – It would be necessary to indicate the units of measurement along the axes.

Response 6: Due to reasons indicated in Response 5 it is not possible to give the units in Fig. 8 (now Fig. 11).

Response to Reviewer#2 Comments

1. Summary

Thank you for sparing your time for reviewing and making positive comments to improve the manuscript. Please find the detailed responses below and the corresponding revisions/corrections colored in red in the revised manuscript.

2. Questions for General Evaluation

Reviewer’s Evaluation

Response and Revisions

Does the introduction provide sufficient background and include all relevant references?

Can be improved

Improved according to comments

Are all the cited references relevant to the research?

Yes

Is the research design appropriate?

Yes

Are the methods adequately described?

Yes

Are the results clearly presented?

Can be improved

Improved according to comments

Are the conclusions supported by the results?

Yes

3. Point-by-point response to Comments and Suggestions for Authors

Comment 1: Authors must follow the journal guidelines.

Response 1: The author supposes that the reviewer suggests the order of writing recommended by the journal: Introduction, Materials and Methods, Results, Discussion, Conclusions. In the writing of the manuscript I tried to do so (as I used the standard template); however, due to the contents of the manuscript I realized that I could not do so hence abandoned it. Basically, the manuscript has two separate contents: A complete theoretical derivation from “the scratch” and its results and then development of a numerical scheme its detailed description and presentation of test cases. Due to this duality it was not for instance to use a heading “Materials and Methods” then present both theory and numerics under it and the same applies to “Results”. Otherwise I followed the guidelines whenever I could. 

Comment 2: The abstract should be improved, inkling a brief summary of : i) background, ii) methods; iii) results; and iv) conclusions.

Response 2: The abstract is improved and the newly added sentences are colored red in the revised manuscript.

Comment 3: The introduction section should be improved. A final paragraph outlining the remaining content of the manuscript should also be included.

Response 3: Suggested improvements together with similar requirements of Reviewer#5 for introduction are done and colored red in the revised manuscript. Lines 42-45, Lines 65-67, Lines 70-73, and Lines 79-81.

Comment 4: Figures 7 and 9 vertical and horizontal axes labels should be coherent.

Response 4: Axes labels of Figs. 7 and 9 (now Figs. 10 and 12) are improved. Reviewer#5 made a similar request for Fig. 7 (now Fig. 10).

Response to Reviewer#3 Comments

1. Summary

Thank you for sparing your time for reviewing and making positive comments to improve the manuscript. Please find the detailed responses below and the corresponding revisions/corrections colored in red in the revised manuscript.

Comments and Suggestions to Author

Reviewer: Nice job, and nice paper.

Author: Thank you.

2. Questions for General Evaluation

Reviewer’s Evaluation

Response and Revisions

Does the introduction provide sufficient background and include all relevant references?

Yes

Are all the cited references relevant to the research?

Yes

Is the research design appropriate?

Yes

Are the methods adequately described?

Yes

Are the results clearly presented?

Yes

Are the conclusions supported by the results?

Yes

Response to Reviewer#4 Comments

1. Summary

Thank you for sparing your time for reviewing and making positive comments to improve the manuscript. Please find the detailed responses below and the corresponding revisions/corrections colored in red in the revised manuscript.

Comments and Suggestions to Author

Reviewer: Well written paper, with robust contents. Authors are kindly invited to create and include a plot to compare the dispersive properties of the novel set of equations with the existing ones (Witting, Peregrine, Nwogu, etc.).

Author: Thank you. A new figure, now Fig. 1 in the revised manuscript is added and necessary explanations colored red are written.

2. Questions for General Evaluation

Reviewer’s Evaluation

Response and Revisions

Does the introduction provide sufficient background and include all relevant references?

Yes

Are all the cited references relevant to the research?

Can be improved

Improved according to comments

Is the research design appropriate?

Yes

Are the methods adequately described?

Yes

Are the results clearly presented?

Yes

Are the conclusions supported by the results?

Yes

3. Point-by-point response to Comments and Suggestions for Authors

Comment 1: Please, add the following references for wave breaking (lines 42-43):

Response 1: The suggested references are added to the introduction together with improvements as suggested in similar comments of Reviewer#2 and Reviewer#5.

Response to Reviewer#5 Comments

1. Summary

Thank you for sparing your time for reviewing and making positive comments to improve the manuscript. Please find the detailed responses below and the corresponding revisions/corrections colored in red in the revised manuscript.

2. Questions for General Evaluation

Reviewer’s Evaluation

Response and Revisions

Does the introduction provide sufficient background and include all relevant references?

Must be improved

Improved according to comments

Are all the cited references relevant to the research?

Yes

Is the research design appropriate?

Yes

Are the methods adequately described?

Must be improved

Improved according to comments

Are the results clearly presented?

Must be improved

Improved according to comments

Are the conclusions supported by the results?

Can be improved

Improved according to comments

3. Point-by-point response to Comments and Suggestions for Authors

Comment 1: Lines 46-66; Please explain the different points of your new scheme in this study more clearly by comparing the method or scheme of previous studies.

Response 1: An explanation making clear the advantages of the new set of equations is added to the text (Lines 65-67 now). More explanations concerning the numerical scheme are added to the text (Lines 70-73 and 79-81 now). Reviewer#2 made a similar comment for improving the introduction.

Comment 2: Lines 42-43; This is an important aspect. Please explain in more detail.

Response 2: Further explanations are added to the text (Lines 42-45 now).

Comment 3: Section 2; Please explain how you handle wave breaking in this section.

Response 3: Wave breaking is not considered in this work; therefore, no explanations are given in this respect.

Comment 4: Figure 6, 8; Please put a scale and color bars.

Response 4: Both Figs. 6 and 8 (now Figs. 9 and 11) are drawn for visual purposes only and both are composed of two different perspective figures (waves and bottom topography) with entirely different scales. Therefore, it is not possible to use true axes (especially in the vertical). Also, since there are two separate figures in one the horizontal axes must be drawn both for waves and bottom topography. Reviewer#1 made a similar comment and asked for axes with labels and dimensions. Due to reasons indicated above this is not possible either. For clarifying this point the following sentence is added to the captions of these figures: “Axes omitted due to different vertical scales for bathymetry and wave field.”

Comment 5: Figure 7; Please put the unit of the horizontal axis. In the caption, please explain why there are two lines in the figure of T=1s and three lines in the figures of T=2 and 3s. And, please explain the reason of discrepancies between calculated and observed values in cases of T=2 and 3 s.

Response 5: Axes labels of Fig. 7 (now Fig. 10) are improved. Reviewer#2 made a similar request for Figs. 7 and 9 (now Figs. 10 and 12). Part concerning the interpretation of results for the topographical lens is completely re-written (Lines 455-462) to explain discrepancies.

Comment 6: Limitations of Boussinesq-type wave equations are discussed in the following article. Please refer to this.

Watanabe, M., Kan, H., Toguchi, K., Nakashima, Y., Roeber, V., Arikawa, T., 2023. Effect of the structural complexity of a coral reef on wave propagation: a case study from Komaka Island, Japan. Ocean Engineering, 287(1), 115632.

Response 6: The indicated reference is included.

1. Summary

Thank you for sparing your time for reviewing and making positive comments to improve the manuscript. Please find the detailed responses below and the corresponding revisions/corrections colored in red in the revised manuscript.

Comments and Suggestions to Author

Reviewer: After discussing the suggestions and comments made by the reviewers and adding to the text and figures, the article became clearer.

However, the main point remained unanswered. I believe that in the text of the article it is necessary to describe in detail how exactly the parameter u0 is calculated - the velocity at the horizon z = 0 (line 102 of the paper). From a physical point of view, this parameter cannot be used, since during the half of the period of the wave there is no water at the horizon z=0 and it cannot have any velocity. This remark was in my first review, but the author got away with the answer that “everyone does it,” which is unacceptable and will be absolutely incomprehensible to readers.

Author: The author is quite disappointed with the remark that he is trying to “get away with” the question by saying that “everyone does it”. Over thirty years of my career I have not tried to get away with anything; on the contrary, I believe it is very rare to find research as clearly explained as mine and it applies to the present work too.

When it comes to the question; yes, in this field everyone, beginning with Sir G. G. Stokes over 150 years ago, uses the variable(s) (velocity potential or equivalently velocity itself) evaluated at the still water level z=0 instead of the actual free surface if he/she is to carry out an analytical formulation whether in the form of a set wave equations or an analytical solution given for a definite case as in Stokes’s well-known second-order theory. The reason to this usage is obvious: the surface displacement itself, which appears in the free surface boundary conditions (kinematic and dynamic) is an unknown and must be solved; therefore, any other kinematic variable (velocity) at the actual free surface cannot be used because it cannot be analytically expressed at this unknown free surface. (Such things can only be done by purely computational techniques, which are necessarily iterative.) Instead, there must be a “known” or a “definite” reference level such as z=0 or z=-h, etc. for which the kinematic variable itself can be formulated as an analytic function to be used to express its value at the “actual” or “perturbed” surface by substituting z=zeta to the formulation. See the attached photos taken from the books by C. C. Mei (The Applied Dynamics of Ocean Surface Waves) and T. Sarpkaya and M. Isaacson (Mechanics of Wave Forces on Offshore Structures) and please especially note the zero subscripts (z=0 level) used for the velocity potential and velocities in the second and third pages. More importantly, note that none of these books (just like all the others) bother to justify the use of variables at z=0. They just introduce it via Taylor series and use it. Does it mean they just “get away with” it? Definitely no, this is a tacitly agreed approach which is accepted a priori without any argument. True, we cannot physically point out a velocity at z=0 during the passage of a wave trough but that is immaterial because what we are doing is we are using a mathematical function at z=0 that enables us to calculate all the velocities with reference to this “real” or “fictitious” velocity at z=0. There may or there may not be a physical velocity present at z=0 at any given instant; but we are always calculating one (from the momentum and continuity eqns.) and using it for our purposes as a convenient (and may be only) way of evaluating all the parameters we need. If we set z=+zeta in eqn (5) of the present work under consideration we get the velocity at wave crest in terms of u0. If we set z=-zeta in the same equation we simply get the velocity at wave trough and that is all to it; this is entirely a mathematical procedure. We have a velocity computed (from continuity and momentum) at a reference level (z=0) and by using it we can “extrapolate” to get any velocity we want. We can do the same things by using eqn (13) where the velocity at an arbitrary depth is used. In this case we can only comfort ourselves by noting that there exists a physical velocity at that depth under the wave trough; but that is all. From the mathematical point of view there is no difference whether we actually mark a definite velocity or not because the only difference between the two velocities is a coordinate transformation from z=0 to z=-αh.

Unquestionably, the ultimate application of the perturbation approach is linear theory. In linear theory we set z=0 everywhere and solve the Laplace equation in a perfectly rectangular domain bounded above by z=0 and below z=-h, subject to the free surface boundary conditions evaluated at z=0 and z=-h (see the first photo). Where do the waves come into picture then? We simply impose a sinusoidal free surface and never talk about the legitimacy of velocity potential or velocities completely formulated from equations evaluated at z=0 and z=-h.

Due to all the above lengthy “philosophical” arguments needed to explain this point, I gave a rather short description in my previous reply: It would neither be possible to enter into such discussion in the manuscript nor has it ever been done in any relevant literature (book or paper) to the best of my knowledge. Simply because it is taken for granted that we can use the velocity at z=0, just like, for instance, we can use the kinematic boundary condition without making a “justification” of using it. (As far as I know the kinematic boundary condition was introduced at a later time to the water wave problem, which had been unsettled till then, in order to have a “complete” solution that brought in the dispersion relationship.)

Nevertheless, as the reviewer insists much on this matter, I included the following justification to line 103 of the manuscript:

“… at the undisturbed still water level z=0. Although used conventionally without questioning, the use of velocities at z=0 may be perplexing from physical point of view when wave troughs are considered. This however does not pose a real problem as the introduction of these velocities is an entirely mathematical process of perturbation approach and actual physical existence of these velocities is immaterial.”

2. Questions for General Evaluation

Reviewer’s Evaluation

Response and Revisions

Does the introduction provide sufficient background and include all relevant references?

Yes

Are all the cited references relevant to the research?

Yes

Is the research design appropriate?

Yes

Are the methods adequately described?

Must be improved

Improved according to comments

Are the results clearly presented?

Yes

Are the conclusions supported by the results?

Yes