Analysis of the reflection and transmission of a nonlinear wave

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Concluding remarks about the one dimensional model

The model presented in this section revealed some important hypothesis that should be considered
to correctly solve the problem of propagation of nonlinear waves.
Firstly, the nonlinear conditions of the problem are key parameters. In this case, the nonlinear parameter is the stepness ka, which should be a small number (ka ≪ 1). The power series expansion only makes physical sense when this parameter is sufficiently small.
Secondly, the boundary conditions at the infinity should be well defined in order to have a wellposed
problem. The radiation conditions of the free waves should be imposed from the statement of the problem. Examples of nonlinear developments can be found in Hammack and Henderson [40], Hsu
et al. [43] and Madsen and Sorensen [49].
The second order problem is a non homogeneous partial differential equation, which has a particular and a homogeneous solution. We observed that the particular solutions should be solved first. This solution fixes the frequency that is applied to the homogeneous solution (in this case 2ω).
To conclude, we consider that the toy model problem presented in this section represents a good exercise for solving a nonlinear wave problem, as a previous step before solving the same problem in the context of surface water waves, which will be analyzed in the next section.

On the convergence of the second order problem

Although the first order problem has a good convergence with respect to the truncation of the series (thus, we do not present any analysis at this order), the second order shows problems with the convergence when we increase the number of vertical modes in the series. The second order solves the problem of finding the proper free wave coefficients that compensate the bound wave field in order to obtain a continuous solution. To do so, the bound wave field should have a bounded amplitude, i.e. it should not grow with the number of modes considered in the problem. The bound wave field is the result of the sum of all the evanescent modes obtained from equations (2.88) and (2.93). As example, we recall the bound wave coefficients Cn,n from equation (2.88).

Convergence of the solution with the number of modes

In this section, the truncation parameter α was fixed at 0.5, because it gives the fastest convergence in the range α ≤ 0.5. We considered in this part, the following configuration: h = 0.3 m, hs = 0.15 m, a = 0.023 m and ω = 2π [1/s]. With these conditions, we calculated the first and second order problems, analyzing the convergence as a function of the number of modes, which is indicated in each graph with the number of modes in the deep water region (N).
In this way, the calculations in the following figures display the improvement of the mode matching when we increase the number of modes. The first result is the first order potential φ1 shown in figures 2.17 and 2.18. These fields permit to observe the fast convergence of the mode matching, giving a negligible error beyond 20 modes. Regarding the vertical profile of both regions along the matching line x = 0, we show in figure, 2.18, the profiles of the velocity potential φ1(x = 0, z). This profile gives a more detailed view of the reduction of the error as a function of the number of modes.

Concluding remarks and perspectives

We have re-visited the multi-modal model proposed by Massel [51]. We confirmed the capability of the model to obtain efficiently the transmission and reflection coefficients, with good accuracy at the first and second order. We have reproduced the calculations published by Massel [51] and Rhee [73], finding good agreement with these references. Of particular interest are the invariance of the transmission phase shift and the invariance of the reflection absolute value, when we compared the problem of a wave going from the deep to the shallow region or vice versa. This invariance could be useful in problems of resonant modes in closed tank with submerged bodies, where the phase shift in both directions is required. We will use these properties later in this work.
We analyzed the convergence of the second order problem, which is not straightforward with a large number of modes. We suppose that the calculations performed by Massel [51] did not have convergence problems by means of the small amount of modes considered by the author. However, nowadays the computational power reveals easily the slow convergence of the first order coefficients (Rn, Tm), which do not compensate the fast growth of the bound wave coefficients Cn,n′ . We propose a criterion for truncating the first order series and input it in the second order problem. The selected threshold (α ≤ 0.5) ensures the convergence of the second order, and renders the solution more accurate as a function of the number of modes. We can note that the model could be improved with the incorporation of the surface tension effect, in order to better model the small scale phenomena.

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Table of contents :

Remerciements
Abstract
1 Introduction 
1.1 Motivation
1.2 Surface wave theory
1.3 Expansion of surface wave equations
1.4 The step problem
1.5 Space-time resolved measurements for water waves
2 Multi-modal model of a nonlinear wave 
2.1 Introduction
2.2 A toy model: One-dimensional nonlinear wave
2.2.1 First order problem
2.2.2 Second order problem
2.2.3 Numeric example
2.2.4 Concluding remarks about the one dimensional model
2.3 Analysis of the reflection and transmission of a nonlinear wave
2.3.1 Statement of the problem
2.3.2 First order problem
2.3.3 Second order problem
2.3.4 Surface elevation of free and bound waves
2.3.5 Numerical example
2.3.6 On the convergence of the second order problem
2.3.7 Convergence of the solution with the number of modes
2.4 Concluding remarks and perspectives
2.4.1 Third order problem
3 Experimental measurements of nonlinear waves 
3.1 Introduction
3.2 Governing equations
3.2.1 The weakly nonlinear model
3.2.2 Surface tension in water waves
3.3 Experimental Set-up
3.4 Results
3.4.1 Waves celerity in the space-time plane
3.4.2 Analysis of the frequency-wavenumber spectra
3.4.3 Separation of free and bound waves
3.4.4 Beating length and the influence of the surface tension
3.4.5 Comparison between theoretical and experimental harmonic modulation
3.5 Concluding remarks
3.6 Supplementary results
3.6.1 Separation of left and right going waves
3.6.2 Complex fit of free and bound waves
3.6.3 Contribution of the terms added to the multi-modal model
4 Low-frequency modes in a tank with submerged step 
4.1 Introduction
4.1.1 Governing equations
4.2 Experimental Set-up
4.3 Characterization of quasi-periodic and purely harmonic regimes
4.3.1 Space-time measurements
4.3.2 Experimental spectrum and phase plane
4.3.3 Transient signal
4.3.4 Low-frequency mode profile
4.4 Low-frequency resonance by varying the forcing frequency
4.5 Low-frequency resonance by forcing amplitude
4.6 Concluding remarks
5 Measurement of attenuation in shallow water 
5.1 Introduction
5.2 Experimental set-up
5.2.1 Geometry and method of measurement
5.2.2 Forcing signal
5.3 Wave packet in multiple directions
5.3.1 Method of measurement of attenuation
5.4 Harmonic one-directional waves
5.5 Theoretical attenuation and comparison with experiments
5.6 Conclusion
6 Wetting properties in small scale experiments 
6.1 Introduction
6.2 Influence of meniscus in an absorbing beach
6.2.1 Experimental measurement of the beach absorption
6.3 Influence of meniscus in a narrow channel
6.3.1 Decomposition in transverse modes
6.3.2 Cut-off frequency
6.3.3 Vertical displacement of a transverse profile
6.3.4 FTP measurement of a waveguide with and without mesh
6.4 Conclusion
7 Experimental measurements of perfect wave absorption 
7.1 Introduction
7.1.1 The radiation damping
7.1.2 Wave absorption
7.1.3 Trapped modes resonance
7.2 Experimental set-up
7.3 Results analysis
7.3.1 Mono-modal propagation
7.3.2 Obtaining the absorption coefficient
7.3.3 Comparison of low and high absorption cases
7.3.4 Variation of resonator parameters
7.4 Conclusions and perspectives
8 Conclusion and perspectives 
8.1 Conclusions of main results
8.2 Perspectives

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