Laboratory and numerical experiments on stem waves due to monochromatic waves along a vertical wall

Yoon, Sung Bum; Lee, Jong-In; Kim, Young-Take; Shin, Choong Hun

doi:https://doi.org/10.5194/npg-25-521-2018

Articles | Volume 25, issue 3

https://doi.org/10.5194/npg-25-521-2018

© Author(s) 2018. This work is distributed under
the Creative Commons Attribution 4.0 License.

https://doi.org/10.5194/npg-25-521-2018

© Author(s) 2018. This work is distributed under
the Creative Commons Attribution 4.0 License.

Articles | Volume 25, issue 3

Research article

|

27 Jul 2018

Research article |

| 27 Jul 2018

Laboratory and numerical experiments on stem waves due to monochromatic waves along a vertical wall

Sung Bum Yoon, Jong-In Lee, Young-Take Kim, and Choong Hun Shin

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Final revised paper (published on 27 Jul 2018)
Supplement to the final revised paper
Preprint (discussion started on 01 Aug 2017)

Interactive discussion

Status: closed

AC: Author comment | RC: Referee comment | SC: Short comment | EC: Editor comment

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- Supplement

RC1: 'Review of npg-2017-35', Anonymous Referee #1, 22 Aug 2017
- AC1: 'Answers to the Referee # 1 comments', Choong Hun Shin, 23 Oct 2017
RC2: 'npg-2017-35', Anonymous Referee #2, 09 Oct 2017
- AC2: 'Answers to the Referee # 2 comments', Choong Hun Shin, 23 Oct 2017
EC1: 'Editor decision', Victor Shrira, 11 Oct 2017
- AC3: 'Answers to the editor comments', Choong Hun Shin, 23 Oct 2017

Peer-review completion

AR: Author's response | RR: Referee report | ED: Editor decision

AR by Choong Hun Shin on behalf of the Authors (23 Oct 2017) Manuscript

ED: Referee Nomination & Report Request started (02 Nov 2017) by Victor Shrira

RR by Anonymous Referee #1 (13 Nov 2017)

RR by Tarmo Soomere (08 Dec 2017)

Suggestions for revision or reasons for rejection

The paper provides an interesting and rich in content experimental study of stem waves developed along a breakwater by trains of originally sinusoidal waves. The authors manage to reproduce basically all theoretically predicted features of the resulting stem waves. The analysis is sound and the interpretation of results is adequate. The revised version takes into account all critics and recommendations by two referees. The use of English is fine and I recommend the manuscript for publication, with some very minor editorial adjustments as indicated below.

Having said that, I wonder whether the authors would consider possible to make a little bit of extra work. As the authors correctly discuss, the definition of stem width (=length of the high common crest of the incoming and reflected wave) is controversial and used in different meanings by different authors. The same problem becomes evident in the limiting case of stationary interactions of shallow-water Kadomtsev-Petviashvili solitons (e.g., Peterson et al. 2003. Soliton interaction as a possible model for extreme waves in shallow water, Nonlinear Processes in Geophysics, 10, 6, 503–510). In this specific case the height of the joint crest varies along the stem, except for the near-resonance case, and it takes time to form a stem of reasonable length (Li et al. 2011. On the Mach reflection of a solitary wave: revisited. Journal of Fluid Mechanics, 672, 326-357).
Even though the stem formation from wave trains considered by the authors is time-dependent and thus very much different from the process of the formation of stationary pattern of interaction of shallow-water solitons, the existence of simple expressions for the core quantities for solitons interactions (e.g. Soomere and Engelbrecht 2005. Extreme elevations and slopes of interacting solitons in shallow water, Wave Motion, 41, 2, 179–192) may put the results in a wider context and can possibly make the results applicable for Mach reflection of solitons as well.
Namely, a rough estimate of the critical angle for resonance of solitons of equivalent amplitude (that match the amplitudes of the incident and reflected waves), crossing angle of the two wave systems and water depth; see, e.g., Soomere 2004. Interaction of Kadomtsev-Petviashvili solitons with unequal amplitudes, Physics Letters A, 332, 1-2, 74–81) might provide some additional explanation why stem formation only occurs for quite a selected set of generated wave fields. I guess that the resonance angle varies considerably for different generated wave heights and thus its value has some potential to clarify why in some cases the stem exists and why it is not present in some other cases. Trains of longer and/or higher waves are in this sense closer to similar trains of shallow-water line solitons and thus the estimates for parameters of soliton interactions should better match the observed development of stem.
However, as this possible amendment would eventually involve references to my own papers, please consider this suggestion as a very gentle one, and in no way as a condition for the acceptance of the manuscript.

Minor suggestions:
Please add the generated wave heights and periods to the figure captions (just to make reading easier).
Abstract, line 19 and page 13, line 12: replace „the lengthening of wave length“ by „the increase in the wave length“
Page 7, line 22: it would be better to say that the same result „apparently“ could be obtained.
Line 8: I agree that „However, it is not an easy task to locate the edge of the flat region.“ Here, again, a reference (even though not 100% relevant) to the case of interacting line solitons (or solitons reflecting from the wall) would make this explanation clearer.
Table 1: move water depth (0.25 m) into the caption as otherwise it creates an empty column.

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RR by Julien Touboul (07 Jan 2018)

RR by Anonymous Referee #5 (14 Jan 2018)

Suggestions for revision or reasons for rejection

Major remarks

While the research presented in the paper is of definite interest to NPG, the following remarks arise:
1) The theory could be presented much more clearly even though these are published, fairly old results. Please clarify and expand.
2) What is missing is a comparison with recent work (including references) on nonlinear stem waves in KP and higher-order water wave approximations than KP; KP and these other equations also allow monochromatic/harmonic standing wave solutions, maybe as solitary waves, which become harmonic waves in the small-amplitude limit. In these cases the amplification is a lot larger (up till 4x) and I miss a discussion of the relevance of these equations and solutions, the single soliton but also harmonic, solitary-wave solutions to these equations, which must somehow be connected with the work presented.
Please update and clarify.
3) There are a lot of figures; are these all required? The nonlinear results with stem waves are the most interesting but I miss in these figures an indication what the extent of the stem wave is, as in Fig. 2. What are the observed stem-wave angles of the wall? There should be some reordering here, with perhaps some results relegated to an appendix or online-only appendix. It would also be useful to mention the values of K in the relevant captions.
Please clarify.
4) What is the relevance to real-world situations. What range of nonlinearities do we expect in these real-world cases? Are the experiments lying in this range? Are the solitary waves lying in this range? Please clarify.

My advice based on the above remarks is therefore to request major modifications.

Minor Remarks
Abstract Line 45: is the word "decrease" correct? Should it not be "increase"? Counterintuitive.
Line 51: Mention relevance to harbours and such.
Line 109 page 4: Overview of reflection of solitary wave-wall-interactions are missing, with the maximum stem wave amplification being 4 for a critical angle in KP -see works of, e.g., Kodama, Yeh and Kodama, Ablowitz and Curtis, Gidel et al., etc., also with respect to the amplification in other water-wave model-approximations of potential flow. This should include the comparison between KP and other models and experiments.
Line 136: equations should be in italics.
Line 143: singular dispersion.
Page 5, section: clarification would be useful, 20 years after these old publications. The REF/DIF manual and reference is also not particularly clear.
Everywhere: formulas need punctuation, also in NPG.
Line 188: explain/define the linear equation set which this linear solution solves.
Line 244: zero-crossings method: please explain. Which zero? What crossing?
Line 293: remove first "as".
Page 9: Over the top amount of detail in these figures. Is there a more compact way to convey this?
Line 319/320: "strong indication of stem wave development": please indicate why this statement holds: arrow in figure, etc; it was not very clear to me;
What is the stem-wave line, i.e. the measured dashed line of Fig. 2; per position plot along the wall and normal to the wall indicate where this dashed line is for this position. I.e. indicate where the stem wave end and what its angle is. Somehow, this stem-wave angle should be available from 2D-horizontal measurements or photographs?
Line 322: I would agree with this statement but which stem-wave angle and position of the end of the stem-wave do we measure or expect? Please add.
Line 354: "stem wave appear clearly"; please indicate where and in which figure (i.e. it is not very clear); add an arrow and symbol to indicate where the stem wave ends. What is the stem-wave angle (measured) for these cases?
Line 355: remove the 2nd "the".
Line 372: explain why; say "because it is linear". [the analytical solution]
Line 373: where does the conclusion come from; which figures? is it true?
I can't really see it also because per figure it is not clearly indicated which one contains a stem wave. Mark this more properly and add reference to the relevant figures or subfigures backing up this statement. What about solutions to KP? Would they be better? Or is Benney-Luke or potential flow required? Please comment. Formula (12): can this be explained/derived quickly; why are shocks expected and is this representation relevant?
Line 413: remove first comma.
Line 432 and Fig 24: I find the multiple lines displayed confusing and the figure caption unclear; there also seems to be only one theta=20 measurement; please clarify the figure and text.
Line 482: rewrite this sentence. Grammar.
Line 483: undulations.
Line 484: an undulation.
Line 487: I don't understand this statement; please clarify.
Line 488: this statement is not true as for larger waves the linear solution does not hold very well. Please amend.
Line 494: indicate in the figures what the values of K are so this is more easy to judge.
Fig. 3: what is the signal imposed on the wavemaker; in order for the results to be reproducible?
Fig. 12: How does this match the sketch in Fig. 2; if the measurement is normal to the wall, where is the dashed line supposed to be, e.g. indicate with a vertical dashed line or cross? Please indicate.
Figures 15 & 18: Again, indicate the stem-wave end-point expected/ measured; cf. the dashed line in Fig. 2 at the appropriate x-location.
Figure 22 for K<0.5: What happens here? Please explain.

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ED: Reconsider after major revisions (further review by editor and referees) (19 Jan 2018) by Victor Shrira

AR by Choong Hun Shin on behalf of the Authors (28 Mar 2018) Author's response Manuscript

ED: Referee Nomination & Report Request started (03 Apr 2018) by Victor Shrira

RR by Julien Touboul (10 Apr 2018)

RR by Onno Bokhove (02 May 2018)

Suggestions for revision or reasons for rejection

I recommend to accept the paper with minor corrections. I think the authors did a great job in trying to accommodate our critique. Yet there are still a few issues left:
(i) The text throughout requires some grammatical corrections pertaining to (better) understanding of statements. I have supplied a pdf with all my comments in confidence (not meant for online publication).
(iii) It is still unclear to me how lambda_s is determined at x=6L and 15L; is alpha determined and then knowing x, lambda_s or is lambda_s visually determined as in the appendix figures seems possible and is indicated? It is the chicken-and-the-egg question.
(iv) The question on the relevance to real-world cases should be answered in the paper, by combining the two figures in the response (figure 4 and the one before that).
(v) How is K calculated from (8) for the measurements and clarify how it is used to find the results in figs 11 and 12 a bit better.
(vi) What is H(x,y,t) as follows from the simulations in terms of the amplitude A calculated from (1)?
(vii) I still don't quite understand the zero crossing method. Please clarify what "next crossing" signifies?
(viii) Figure 5: define axes and state what they indicate; add the real-world swell symbols used in the response as well and say in text and caption that the results seem thus relevant to the real world.
(ix) What is a node line (define please)?

Minor remarks
1) A lot of grammatical glitches have been indicated in the pdf-file.
2) Formula (8): How is k determined in experiment? Are C and Cg then calculated using linear wave theory to find the K's used in figs 9 and 10? If so, say so. h is measured and known, A0 measured and theta0 as well so that K follows.
3) Line 14 on page 4 above section 2.2: What are h(x,y,t) and \phi(x,y,t)
in terms of A? That would be useful as h (and perhaps phi) are used later on.
4) Line 7 page 7: I still don't understand it: what is meant by "next crossing point"? Please clarify.
5) Lines 11-16: I don't think this addition is needed.
6) Line 31 page 7: All? I meant to choose a selective subset of figures for the main text and maybe relegate the rest to the appendix, provided they are useful. But the current setting does work although the subfigures with the stem waves indicated could be part of the main text. Either way would work.
7) Page 8 line 7: That is not a fair comment as it is much easier in the case of a solitary wave to find the Mach stem.
8) Formula (13): I am confused here: the figures in appendix indicate
that lambda_s is determined by inspection and drawing (as the more flattened region), as now indicated in the appropriate cases, which is fine, and subsequently (13) is used, given x, to find alpha. But somewhere in captions it says lambda_s is determined using (13), in which case my question is how alpha has been measured from the data? Please clarify.
9) Page 9 line 31: I am lost here: given x, is lambda_s measured and then alpha calculated or the other way around: alpha is measured in which case please clarify how alpha is measured please clarify issue?
10) Page 11 line 29: slightly? Can you indicate where this conclusion (that REF/DIF simulation angles are slightly different than experimental ones)
comes from as the figures in the appendix indicate to me that experiments and REF/DIF are in agreement and any differences are within measurement error?
11) Line 9 on page 12: provide the link between A and H as said/asked earlier.
12) Line 16 page 12: part of K (namely finding C and Cg) seems to be based on linear theory? Is that fair?
13) Lines 20-21, formulas (17) and (18): I am lost here? Please clarify the comparison. E.g., is gamma supposed to be gamma=tan theta_0/(sqrt(3)*cos theta0)? If so then say so. How was gamma determined?

Referee Report: PDF

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ED: Publish subject to minor revisions (review by editor) (03 May 2018) by Victor Shrira

AR by Choong Hun Shin on behalf of the Authors (25 May 2018) Author's response Manuscript

ED: Publish subject to technical corrections (10 Jun 2018) by Victor Shrira

ED: Publish subject to technical corrections (11 Jun 2018) by Victor Shrira

AR by Choong Hun Shin on behalf of the Authors (19 Jun 2018) Manuscript

Short summary

Laboratory and numerical experiments are conducted to investigate stem waves due to incidence of monochromatic waves. For larger-amplitude waves with smaller angle of incidence, the measured data clearly show stem waves. The resonant interactions between the incident and reflected waves predicted for solitary waves are not observed for the periodic Stokes waves. The existence and the properties of stem waves found theoretically via simulations are favorably supported by the physical experiments.