Accuracy Examination of the Fourier Series Approximation for Almost Limiting Gravity Waves on Deep Water

Round 1

Reviewer 1 Report

Comments for author File: Comments.pdf

Author Response

Reply to the reviewers’ comments for the draft (series no.2645787). When revising the manuscript, we mark the corrections in red.

Review #1:

This manuscript employs the Fourier series approximation method proposed by Rienecker and Fenton (1981) to solve the gravity wave equation in deep water. The results agree with the published data. Additionally, the authors find that this algorithm fails at the highest wave because it cannot simultaneously satisfy both kinematic and dynamic boundary conditions. Moreover, the authors conjecture that the physical quantities of gravity waves may not reach their peak values before breaking. However, these conclusions require further elaboration for the following reasons:

1. The authors use the Fourier series approximation to convert the governing equations (2)-(3) into a set of algebraic equations and employ Newton’s method to solve these equations. The divergence observed when N > 40 or, in the case of highest wave, may be attributed to Newton’s method rather than the Fourier series approximation. Recent studies have successfully applied the Fourier series approximation with thousands of truncation terms for limiting Stokes’ wave.

Reply: We add some literature and introduce the methods and results of using hundreds or even thousands of terms in the Fourier series. Explain the reason why FSA cannot reach 40 terms.

Action: (Line 495-519)

Unlike previous studies that were based on the complex potential plane, the formulation of FSA is based on the physical plane. The boundary conditions on the free surface, Equations (6) and (7), contribute to FSA in an implicit form of x and η(x), which is unknown and changes with x. Therefore, the coefficient of the cos term of FSA is not simply an unknown value but is related to x. These unknown variables in FSA can be solved using KFSBC and DFSBC at the same time. Zhao et al. [47] pointed out that using different series terms to represent the wave profile and the velocity potential can simultaneously reduce the residuals of KFSBC and DFSBC for large waves. The improvement in KFSBC is quite significant compared to DFSBC. Because the wave profile is assumed to be a Fourier cosine series, the slope always remains zero at the crest, so that the residual values of KFSBC and DFSBC are always zero at the crest. However, the residual values of KFSBC and DFSBC at different positions vibrate and attenuate from the crest to the trough, and the maximum value occurs at the point adjacent to the crest (see Figure 8 of Zhao et al. [47]). Lukomsky et al. [63] used the Fourier series with hundreds of terms to represent the velocity potential and the wave profile of which the harmonics are obtained by using kinematic and dynamic boundary conditions alone recursively. The wave profile is not assumed to be a Fourier cosine series in FSA, which is the main difference from Zhao et al. [47] and Lukomsky et al. [63].

The possible Fourier solution on the complex plane is in an explicit form of only potential; additionally, the value of the conjugate stream function at the surface is set to zero. While all coefficients of the Fourier series are unknown constants which can be solved by using the DFSBC condition. Zhong and Liao [43] obtained all coefficients of even truncated thousands of terms of a Fourier series for the limiting wave on deep water using HAM. In terms of methodology, the solution of FSA on the physical plane will be more difficult than the Fourier series solution on the complex variable plane.

2. Since this manuscript considers only a small number of truncation terms N = 32, a careful examination of the reported physical quantities, especially for the highest wave case, is necessary. It is also suggested that the wave profile be reported with different values of N.?

Reply: The influence of different N values ​​on the accuracy of calculating wave physical properties has been explained in Section 3.1 of the original manuscript, especially Table 1. In the section of “introduction”, we add a description of the comparison and verification of the calculated values ​​and experimental values ​​of different N values ​​in the Rienecker and Fenton (1982).

Action: (Line 70-77)

Rienecker and Fenton [1] applied FSA to compute physical quantities varying with position and time, physical quantities characteristic of the wave train. Furthermore, by comparisons with two experiments which measure fluid velocities under the crest in a wave tank, a better agreement between FSA with N=8 and these experiments than other analytical theories indicates the physical validity of FSA and the results for N = 16 and 32 were indistinguishable from the result for N=8. Rienecker and Fenton [1] described the FSA algorithm for wave shoaling and compared with the experiments of Hansen and Svendsen [44]. The results show quite good agreement.

3. The Fourier series approximation and Newton’s method are not new, and a small truncation number is used. The reasoning behind the conjecture, ”the physical quantities of gravity wave may not reach their peak values before breaking,” is not sufficiently supported.?

Reply: Following the reviewer's opinion, we have deleted the statement of this inference in the abstract and conclusion.

Author Response File: Author Response.docx

Reviewer 2 Report

Comments for author File: Comments.pdf

Author Response

Reply to the reviewers’ comments for the draft (series no.2645787). When revising the manuscript, we mark the corrections in red.

Review #2:

The manuscript titled "Accuracy Examination of the Fourier Series Approximation for Almostlimiting Gravity Waves on Deep Water" addresses the evaluation of Fourier Series Approximation (FSA) for almost-limiting gravity waves on deep water. The authors have explored a classical mathematical problem of permanent gravity wave propagation and focused on understanding the accuracy and limitations of FSA, a method not previously evaluated for almost-limiting Stokes waves.

In light of the identified strengths and weaknesses, I recommend a minor revision of the paper. The authors should address the mentioned shortcomings, particularly the need for empirical validation and enhanced readability, to strengthen the paper's contribution to the field.

1. However, the paper also has some weaknesses. Despite the comprehensive theoretical approach, there is a lack of practical or empirical data to support the theoretical findings, which might limit the paper's applicability. The manuscript could benefit from case studies or real-world applications to demonstrate the practical implications of the findings.

Reply: We add a paragraph explaining the physical verification of FSA compared to experiments described in the Rienecker and Fenton [1]. The wave conditions in these experiments were not large waves. This explanation is in lines 70 to 77 of the revised manuscriptAs for the almost-limiting wave test, no one has conducted it yet. It is mainly limited by the wavemaker theory. This explanation is in lines 78 to 89 of the revised manuscript.

Action: (Line: 70-77)

Rienecker and Fenton [1] applied FSA to compute physical quantities varying with position and time, physical quantities characteristic of the wave train. Furthermore, by comparisons with two experiments which measure fluid velocities under the crest in a wave tank, a better agreement between FSA with N=8 and these experiments than other analytical theories indicates the physical validity of FSA and the results for N = 16 and 32 were indistinguishable from the result for N=8. Rienecker and Fenton [1] described the FSA algorithm for wave shoaling and compared with the experiments of Hansen and Svendsen [44]. The results show quite good agreement.

(Line: 78-89)

The motion of a paddle with regular oscillation in a two-dimensional flume can produce progressive waves and a series of standing (evanescent) waves, of which the amplitudes decay exponentially with distance from the wavemaker (Dean and Dalrymple [45]). However, Vivanco et al. [46] proposed the pedal-wavemaking method to generate a regular surface wave excluding evanescent waves. The proper synchronisation of the orbital motion on the bottom can emulate deep-water behaviour according to the linear Airy theory. When the wavemaker moves with large displacements there are significant non-linear effects between these waves and the paddle motion that occur to form the waves of different size and shape at different locations away from the wavemaker. The sharpness of the crest for the very highest and longest waves due to high non-linearity cannot be solved. Therefore, experiments on the kinematic characteristics of large waves on a flat bottom are still unavailable, regardless of the results of almost-limiting waves.

2. Additionally, while the manuscript is rich in mathematical content, it may be overly dense for readers not familiar with the advanced mathematical concepts discussed. Simplifying some sections or providing more intuitive explanations could make the paper more accessible to a broader audience.

Reply: We have deleted the well-known Newton iteration method and only retained the comparison of wave speed and sharpness to shorten the content of the comparison results and strengthen the interpretation and contribution of the results to attract readers' interest as much as possible.

Action: Please refer to the modified part of the revised manuscript.

3. To improve the manuscript's quality, the authors could consider incorporating empirical data or case studies to validate the theoretical findings. Moreover, addressing the practical implications and potential applications of the research would enhance its relevance to real-world problems. It would also be beneficial to discuss the limitations of the study more thoroughly, acknowledging any assumptions made and their impact on the results.

Reply: We add an explanation of why FSA cannot add series terms to accurately calculate large waves, and explain the differences with other previous methods. We add an explanation of why FSA cannot add series terms to accurately calculate large waves, and explain the differences with other previous methods.

Action: (Line: 495-519)

Unlike previous studies that were based on the complex potential plane, the formulation of FSA is based on the physical plane. The boundary conditions on the free surface, Equations (6) and (7), contribute to FSA in an implicit form of x and η(x), which is unknown and changes with x. Therefore, the coefficient of the cos term of FSA is not simply an unknown value but is related to x. These unknown variables in FSA can be solved using KFSBC and DFSBC at the same time. Zhao et al. [47] pointed out that using different series terms to represent the wave profile and the velocity potential can simultaneously reduce the residuals of KFSBC and DFSBC for large waves. The improvement in KFSBC is quite significant compared to DFSBC. Because the wave profile is assumed to be a Fourier cosine series, the slope always remains zero at the crest, so that the residual values of KFSBC and DFSBC are always zero at the crest. However, the residual values of KFSBC and DFSBC at different positions vibrate and attenuate from the crest to the trough, and the maximum value occurs at the point adjacent to the crest (see Figure 8 of Zhao et al. [47]). Lukomsky et al. [63] used the Fourier series with hundreds of terms to represent the velocity potential and the wave profile of which the harmonics are obtained by using kinematic and dynamic boundary conditions alone recursively. The wave profile is not assumed to be a Fourier cosine series in FSA, which is the main difference from Zhao et al. [47] and Lukomsky et al. [63].

The possible Fourier solution on the complex plane is in an explicit form of only potential; additionally, the value of the conjugate stream function at the surface is set to zero. While all coefficients of the Fourier series are unknown constants which can be solved by using the DFSBC condition. Zhong and Liao [43] obtained all coefficients of even truncated thousands of terms of a Fourier series for the limiting wave on deep water using HAM. In terms of methodology, the solution of FSA on the physical plane will be more difficult than the Fourier series solution on the complex variable plane.

4. In the context of the comprehensive literature review and the discussion on the accuracy and applicability of the Fourier Series Approximation for almost-limiting gravity waves, it is pertinent to consider additional relevant studies that could further enhance the manuscript's depth and scope. In particular, the efficient computation of solitary waves to the full Euler equations in gravity and capillary-gravity regimes is discussed in the following previously published studies

Reply: In addition to the four papers suggested by the reviewers, we have added a total of 11 papers to respond to two the reviewers' comments.

Action: (Line: 64-69)

 An efficient and fast computation was developed for pure solitary waves by Clamond and Dutykh [39] and Dutykh and Clamond [40] or for generalised solitary gravity–capillary water waves by Clamond et al. [41] and Dutykh et al. [42]. Zhong and Liao [43] used the homotopy analysis method (HAM) to have convergent computation for the limiting Stokes waves in arbitrary water depth and solitary waves in extremely shallow water.

(Line: 599-602)

Perhaps this problem may be caused by the truncation of finite terms in the original infinite-series solution. Clamond's RCW (renormalized cnoidal wave) method of converting infinite series into a simple Fourier-Padé form may be a good inspiration for improving FSA in the future.

Author Response File: Author Response.docx

Round 2

Reviewer 1 Report

Required revisions have been addressed.

Reviewer 2 Report

I am happy to recommend the acceptance of this manuscript.