# Five-Wave Resonances in Deep Water Gravity Waves: Integrability, Numerical Simulations and Experiments

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## Abstract

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## 1. Introduction

## 2. Deep-Water Surface Waves in One-Dimensional Propagation

#### 2.1. Fifth-Order Hamiltonian Theory in Natural Variables

#### 2.2. The Simplest Arrangement of Five-Wave Resonances

- Explicit formulae for the five-wave resonant manifold, leading to five different types of resonance, numbered from (i) to (v).
- Closed-form expressions for the five-wave interaction coefficients evaluated at the resonant manifold. In particular, it was shown that these interaction coefficients are equal to zero in resonance types (iii) and (iv), and nonzero in the resonance types (i), (ii) and (v).

- First resonant quintet:$${K}_{2}+{K}_{2}+(-{K}_{2})={K}_{3}+(-{K}_{1}),\phantom{\rule{2.em}{0ex}}{K}_{1}:{K}_{2}:{K}_{3}=16:9:25\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{2.em}{0ex}}({K}_{1}+{K}_{2}={K}_{3})$$
- Second resonant quintet:$${K}_{2}+{K}_{2}+(-{K}_{3})={K}_{3}+(-{K}_{4}),\phantom{\rule{2.em}{0ex}}{K}_{2}:{K}_{3}:{K}_{4}=1:3:4\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{2.em}{0ex}}({K}_{2}+{K}_{3}={K}_{4})$$

- Third resonant quintet (vanishing interaction coefficient):$${K}_{2}+{K}_{2}+{K}_{2}={K}_{4}+(-{K}_{2}),\phantom{\rule{2.em}{0ex}}{K}_{2}:{K}_{3}:{K}_{4}=1:3:4\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{2.em}{0ex}}({K}_{2}+{K}_{3}={K}_{4})$$

#### 2.3. Non-Integrability of the Five-Wave Resonance in the Case of Encountering Waves

#### 2.4. Analysis of the Scenario of Encountering Waves in the Small-Steepness Case

## 3. Numerical Simulations of the Fifth-Order Governing Equations in Natural Variables

#### 3.1. Encountering Plane Waves

#### 3.2. Encountering Wave Packets

## 4. Experiments

#### Encountering Wavepacket Experiments

## 5. Conclusions and Discussion

- On the theoretical front, the 5-wave resonant manifold and the interaction coefficients on this resonant manifold had been obtained in a series of papers over two decades ago [4,5,6], triggered by the discovery that the interaction coefficients vanish identically on the 4-wave resonant manifold for one-dimensional propagation of water gravity waves [3]. We used these results to find the simplest 5-wave resonance that can be made out of a triad of wavevectors and their negatives, and calculated its normal-form Hamiltonian. We proved that the system is not integrable, but it lacks just one constant of motion to become integrable, so symmetric scenarios can produce integrable systems.
- On the front of numerical simulations of the governing partial differential equations, the equations had been obtained over five decades ago [15], as a power series in terms of the steepness effectively. The numerical implementation we needed to use in order to accurately resolve 5-wave interactions is the one that uses up to 6-wave interactions in a pseudo-spectral setting [40]. Such an implementation requires a higher-than-usual dealiasing and thus can get quite expensive in terms of the required spectral resolution. We managed to validate our implementation against some benchmarks, and were able to establish the existence of the resonance and quantify its effects in terms of the energy transferred to target modes. We considered encountering plane waves and also encountering wave packets, in a simulated tank that is 300 m long. The plane-wave case provided the most efficient energy transfer in terms of Hamiltonian energy, while the wavepackets case provided a higher efficiency in terms of the probe measurements of surface elevation at a point along the tank.
- On the experimental front the main difficulty is to fine tune the amplitudes and frequencies in order to capture the resonance, but we got this from hindsight. Our preliminary experiments seem to show that the resonance exists physically, although as can be seen in Figure 18, the efficiency is relatively small.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Construction of 5-Wave Resonances Using Only Three Different Positive Wavevectors J_{1}, J_{2}, J_{3} (with 0 < J_{2} < J_{1} and J_{3} = J_{1} + J_{2}) Along with Their Negatives

- (i)
- All wavevectors are positive.
- (ii)
- $p,q>0$, and one of the ${k}_{1},{k}_{2},{k}_{3}$ is negative.
- (iii)
- $p,q>0$, and two of the ${k}_{1},{k}_{2},{k}_{3}$ are negative.
- (iv)
- $p,q$ have different signs, and ${k}_{1},{k}_{2},{k}_{3}>0$.
- (v)
- $p,q$ have different signs, and one of the ${k}_{1},{k}_{2},{k}_{3}$ is negative.

- (i)
- As all wavevectors are positive, we need to restrict ${k}_{1},{k}_{2},{k}_{3},p,q\in \{{J}_{1},{J}_{2},{J}_{3}\}$. As we must impose (A1) we conclude $\{{k}_{1},{k}_{2},{k}_{3}\}\ne \{{J}_{1},{J}_{2},{J}_{3}\}$. Therefore we have two subcases: (i.1) ${k}_{1}={k}_{2}={k}_{3}$ and (i.2) ${k}_{1}={k}_{2}\ne {k}_{3}$. In subcase (i.1) the option $p=q\ne {k}_{1}$ leads to the momentum equation $3{k}_{1}=2p$ but the resonance condition is $3\sqrt{{k}_{1}}=2\sqrt{p}$ which contradicts the momentum equation. The only other option is $p\ne q\ne {k}_{1}\ne p$. We have the following three instances:(i.1.a) ${k}_{1}={k}_{2}={k}_{3}={J}_{2},\phantom{\rule{1.em}{0ex}}p={J}_{1},\phantom{\rule{1.em}{0ex}}q={J}_{3}$. Here, the momentum equation is $3{J}_{2}={J}_{1}+{J}_{3}$ which implies ${J}_{2}={J}_{1}$ and thus ${J}_{3}=2{J}_{2}$, so the resonance condition is $3\sqrt{{J}_{2}}=\sqrt{{J}_{2}}+\sqrt{2{J}_{2}}$, which has no solution.(i.1.b) ${k}_{1}={k}_{2}={k}_{3}={J}_{1}\phantom{\rule{1.em}{0ex}}p={J}_{2},\phantom{\rule{1.em}{0ex}}q={J}_{3}$. Here, the momentum equation is $3{J}_{1}={J}_{2}+{J}_{3}$ which implies ${J}_{1}={J}_{2}$ and thus ${J}_{3}=2{J}_{2}$, so the resonance condition is $3\sqrt{{J}_{2}}=\sqrt{{J}_{2}}+\sqrt{2{J}_{2}}$, which has no solution.(i.1.c) ${k}_{1}={k}_{2}={k}_{3}={J}_{3}\phantom{\rule{1.em}{0ex}}p={J}_{1},\phantom{\rule{1.em}{0ex}}q={J}_{2}$. Here, the momentum equation is $3{J}_{3}={J}_{1}+{J}_{2}$ which implies ${J}_{3}=0$, with no solution.In conclusion subcase (i.1) has no solution. We now study subcase (i.2) ${k}_{1}={k}_{2}\ne {k}_{3}$. In this subcase we must have $p=q\ne {k}_{j}$ for all $j=1,2,3$. We have the following six instances:(i.2.a) ${k}_{1}={k}_{2}={J}_{2},\phantom{\rule{1.em}{0ex}}{k}_{3}={J}_{1},\phantom{\rule{1.em}{0ex}}p=q={J}_{3}$. Here, the momentum equation is $2{J}_{2}+{J}_{1}=2{J}_{3}$ which implies ${J}_{1}=2{J}_{1}$, with no solution.(i.2.b) ${k}_{1}={k}_{2}={J}_{1},\phantom{\rule{1.em}{0ex}}{k}_{3}={J}_{2},\phantom{\rule{1.em}{0ex}}p=q={J}_{3}$. Here, the momentum equation is $2{J}_{1}+{J}_{2}=2{J}_{3}$ which implies ${J}_{2}=2{J}_{2}$, with no solution.(i.2.c) ${k}_{1}={k}_{2}={J}_{2},\phantom{\rule{1.em}{0ex}}{k}_{3}={J}_{3},\phantom{\rule{1.em}{0ex}}p=q={J}_{1}$. Here, the momentum equation is $2{J}_{2}+{J}_{3}=2{J}_{1}$ which implies ${J}_{1}=3{J}_{2}$ and thus ${J}_{3}=4{J}_{2}$, so the resonance condition is $2\sqrt{{J}_{2}}+\sqrt{4{J}_{2}}=2\sqrt{3{J}_{2}}$, which has no solution.(i.2.d) ${k}_{1}={k}_{2}={J}_{1},\phantom{\rule{1.em}{0ex}}{k}_{3}={J}_{3},\phantom{\rule{1.em}{0ex}}p=q={J}_{2}$. Here, the momentum equation is $2{J}_{1}+{J}_{3}=2{J}_{2}$ which implies ${J}_{2}=3{J}_{1}$ which has no solution (because ${J}_{2}\le {J}_{1}$).(i.2.e) ${k}_{1}={k}_{2}={J}_{3},\phantom{\rule{1.em}{0ex}}{k}_{3}={J}_{2},\phantom{\rule{1.em}{0ex}}p=q={J}_{1}$. Here, the momentum equation is $2{J}_{3}+{J}_{2}=2{J}_{1}$ which implies $3{J}_{2}=0$, with no solution.(i.2.f) ${k}_{1}={k}_{2}={J}_{3},\phantom{\rule{1.em}{0ex}}{k}_{3}={J}_{1},\phantom{\rule{1.em}{0ex}}p=q={J}_{2}$. Here, the momentum equation is $2{J}_{3}+{J}_{1}=2{J}_{2}$ which implies $3{J}_{1}=0$, with no solution.In conclusion subcase (i.2) has no solution. Therefore, case (i) has no solution.
- (ii)
- In this case we restrict ${k}_{1},{k}_{2},p,q\in \{{J}_{1},{J}_{2},{J}_{3}\}$ and ${k}_{3}\in \{-{J}_{1},-{J}_{2},-{J}_{3}\}$, with $\{{k}_{1},{k}_{2}\}\cap \{p,q\}=\varnothing $ following from (A1). The momentum condition reads ${k}_{1}+{k}_{2}=\left|{k}_{3}\right|+p+q$. As in case (i) we have two subcases: (ii.1) ${k}_{1}={k}_{2}$ and (ii.2) ${k}_{1}\ne {k}_{2}$. In subcase (ii.1) the option $p=q\ne {k}_{1}$ leads to the momentum condition $2{k}_{1}=\left|{k}_{3}\right|+2p$, leading to $|{k}_{3}|=2({k}_{1}-p)$, so ${k}_{1}>p$. However, the resonance condition reads $2\sqrt{{k}_{1}}+\sqrt{|{k}_{3}|}=2\sqrt{p}$, leading to $\sqrt{|{k}_{3}|}=2(\sqrt{p}-\sqrt{{k}_{1}})$, so $p>{k}_{1}$, a contradiction.The only other option is $p\ne q\ne {k}_{1}\ne p$. We have the following three instances:(ii.1.a) ${k}_{1}={k}_{2}={J}_{2},\phantom{\rule{1.em}{0ex}}p={J}_{1},\phantom{\rule{1.em}{0ex}}q={J}_{3}$. Here, the momentum condition is $2{J}_{2}=\left|{k}_{3}\right|+{J}_{1}+{J}_{3}$ which implies ${J}_{2}=\left|{k}_{3}\right|+2{J}_{1}$, with no solution (because ${J}_{2}\le {J}_{1}$).(ii.1.b) ${k}_{1}={k}_{2}={J}_{1},\phantom{\rule{1.em}{0ex}}p={J}_{2},\phantom{\rule{1.em}{0ex}}q={J}_{3}$. Here, the momentum condition is $2{J}_{1}=\left|{k}_{3}\right|+{J}_{2}+{J}_{3}$ which implies $|{k}_{3}|={J}_{1}-2{J}_{2}$, with only solution $|{k}_{3}|={J}_{2}$, leading to ${J}_{1}=3{J}_{2}$ and thus ${J}_{3}=4{J}_{2}$. Thus, the resonance condition reads $2\sqrt{3{J}_{2}}+\sqrt{{J}_{2}}=\sqrt{{J}_{2}}+\sqrt{4{J}_{2}}$, with no solution.(ii.1.c) ${k}_{1}={k}_{2}={J}_{3},\phantom{\rule{1.em}{0ex}}p={J}_{2},\phantom{\rule{1.em}{0ex}}q={J}_{1}$. Here, the momentum condition is $2{J}_{3}=\left|{k}_{3}\right|+{J}_{2}+{J}_{1}$ which implies $|{k}_{3}|={J}_{3}$. The resonance condition reads $2\sqrt{{J}_{3}}+\sqrt{{J}_{3}}=\sqrt{{J}_{1}}+\sqrt{{J}_{2}}$, or $3\sqrt{{J}_{1}+{J}_{2}}=\sqrt{{J}_{1}}+\sqrt{{J}_{2}}$. Squaring this gives $9({J}_{1}+{J}_{2})={J}_{1}+{J}_{2}+2\sqrt{{J}_{1}{J}_{2}}$, thus ${J}_{1}+{J}_{2}=\sqrt{{J}_{1}{J}_{2}}/4$. Squaring again gives ${J}_{1}^{2}+{J}_{2}^{2}+2{J}_{1}{J}_{2}={J}_{1}{J}_{2}/16$, with no real solution.In conclusion subcase (ii.1) has no solution. We now study subcase (ii.2) ${k}_{1}\ne {k}_{2}$. In this subcase we must have $p=q\ne {k}_{j}$ for all $j=1,2,3$. We have the following three instances:(ii.2.a) ${k}_{1}={J}_{2},\phantom{\rule{1.em}{0ex}}{k}_{2}={J}_{1},\phantom{\rule{1.em}{0ex}}p=q={J}_{3}$. Here, the momentum equation is ${J}_{2}+{J}_{1}=\left|{k}_{3}\right|+2{J}_{3}$, which implies $0=|{k}_{3}|+{J}_{3}$, with no solution.(ii.2.b) ${k}_{1}={J}_{2},\phantom{\rule{1.em}{0ex}}{k}_{2}={J}_{3},\phantom{\rule{1.em}{0ex}}p=q={J}_{1}$. Here, the momentum equation is ${J}_{2}+{J}_{3}=\left|{k}_{3}\right|+2{J}_{1}$, which implies $|{k}_{3}|=2{J}_{2}-{J}_{1}$, so ${J}_{1}<2{J}_{2}$. As $|{k}_{3}|\in \{{J}_{1},{J}_{2},{J}_{3}\}$, we can check that $|{k}_{3}|={J}_{3}$ is not possible as it implies ${J}_{2}=2{J}_{1}<4{J}_{2}$, a contradiction. We can check that the two remaining choices $|{k}_{3}|={J}_{1}$ or $|{k}_{3}|={J}_{2}$ imply ${J}_{1}={J}_{2}$. So we conclude $|{k}_{3}|={J}_{1}={J}_{2}$ and thus ${J}_{3}=2{J}_{2}$. Thus, the resonance condition reads $\sqrt{{J}_{2}}+\sqrt{2{J}_{2}}+\sqrt{{J}_{2}}=2\sqrt{{J}_{2}}$, with no solution.(ii.2.c) ${k}_{1}={J}_{1},\phantom{\rule{1.em}{0ex}}{k}_{2}={J}_{3},\phantom{\rule{1.em}{0ex}}p=q={J}_{2}$. Here, the momentum equation is ${J}_{1}+{J}_{3}=\left|{k}_{3}\right|+2{J}_{2}$, which implies $|{k}_{3}|=2{J}_{1}-{J}_{2}$. As $|{k}_{3}|\in \{{J}_{1},{J}_{2},{J}_{3}\}$, we can check that $|{k}_{3}|={J}_{3}$ implies ${J}_{1}=2{J}_{2}$ and $|{k}_{3}|={J}_{3}=3{J}_{2}$, so the resonance condition reads $\sqrt{2{J}_{2}}+\sqrt{3{J}_{2}}+\sqrt{3{J}_{2}}=2\sqrt{{J}_{2}}$, with no solution. The two remaining choices $|{k}_{3}|={J}_{2}$ or $|{k}_{3}|={J}_{1}$ imply $|{k}_{3}|={J}_{1}={J}_{2}$ and ${J}_{3}=2{J}_{2}$. Thus, the resonance condition reads $\sqrt{{J}_{2}}+\sqrt{2{J}_{2}}+\sqrt{{J}_{2}}=2\sqrt{{J}_{2}}$, with no solution.In conclusion subcase (ii.2) has no solution. Therefore, case (ii) has no solution.
- (iii)
- This case has zero interaction coefficients so it will not be considered.
- (iv)
- This case has zero interaction coefficients so it will not be considered.
- (v)
- In this case we restrict ${k}_{1},{k}_{2},p\in \{{J}_{1},{J}_{2},{J}_{3}\}$ and ${k}_{3},q\in \{-{J}_{1},-{J}_{2},-{J}_{3}\}$, with $\{{k}_{1},{k}_{2}\}\cap \left\{p\right\}=\varnothing $ and ${k}_{3}\ne q$ following from (A1). Instead of considering explicitly all 27 possible instances we will use the results from [5] regarding inequalities amongst frequencies. These translate directly to inequalities amongst wavevectors, which in our notation can be summarised as:$${k}_{1}\le {k}_{2}<p,\phantom{\rule{2.em}{0ex}}|{k}_{3}|<|q|,\phantom{\rule{2.em}{0ex}}p\ne \left|q\right|.$$As we assume without loss of generality ${J}_{2}\le {J}_{1}<{J}_{3}$, it follows that $p\ne {J}_{2}$ and $\left|q\right|\ne {J}_{2}$ because ${J}_{2}$ is the smallest wavenumber. Thus, there are two subcases: (v.1) $p={J}_{3},\phantom{\rule{1.em}{0ex}}q=-{J}_{1}$, and (v.2) $p={J}_{1},\phantom{\rule{1.em}{0ex}}q=-{J}_{3}$.In subcase (v.1) $p={J}_{3},\phantom{\rule{1.em}{0ex}}q=-{J}_{1}$, inequalities (A2) imply ${k}_{1},{k}_{2}<{J}_{3}$ and ${k}_{3}=-{J}_{2}$, with the now strict inequality ${J}_{2}<{J}_{1}$. There are three options:(v.1.a) ${k}_{1}={k}_{2}={J}_{1}$. Here the momentum condition reads $2{J}_{1}+(-{J}_{2})={J}_{3}+(-{J}_{1})$, which simplifies to ${J}_{1}={J}_{2}$, a contradiction.(v.1.b) ${k}_{1}={J}_{2},\phantom{\rule{1.em}{0ex}}{k}_{2}={J}_{1}$. Here the momentum condition reads ${J}_{2}+{J}_{1}+(-{J}_{2})={J}_{3}+(-{J}_{1})$, which simplifies to ${J}_{1}={J}_{2}$, a contradiction.(v.1.c) ${k}_{1}={k}_{2}={J}_{2}$. Here the momentum condition reads $2{J}_{2}+(-{J}_{2})={J}_{3}+(-{J}_{1})$, which is identically satisfied (because ${J}_{3}={J}_{2}+{J}_{1}$). We turn to the resonance condition to find $2\sqrt{{J}_{2}}+\sqrt{{J}_{2}}=\sqrt{{J}_{1}+{J}_{2}}+\sqrt{{J}_{1}}$, or $\sqrt{{J}_{1}+{J}_{2}}=3\sqrt{{J}_{2}}-\sqrt{{J}_{1}}$. Squaring this gives ${J}_{1}+{J}_{2}=9{J}_{2}+{J}_{1}-6\sqrt{{J}_{1}{J}_{2}}$, or $3\sqrt{{J}_{1}{J}_{2}}=4{J}_{2}$. Squaring again gives $9{J}_{1}=16{J}_{2}$. Thus ${J}_{1}=16{J}_{2}/9$ and ${J}_{3}=25{J}_{2}/9$. In summary this leads to a 5-wave resonance parameterised by $K\in {\mathbb{Z}}_{+}$ as follows:$$9K+9K+(-9K)=25K+(-16K),\phantom{\rule{2.em}{0ex}}\sqrt{9K}+\sqrt{9K}+\sqrt{9K}=\sqrt{25K}+\sqrt{16K}\phantom{\rule{0.166667em}{0ex}}.$$In subcase (v.2) $p={J}_{1},\phantom{\rule{1.em}{0ex}}q=-{J}_{3}$, inequalities (A2) imply ${k}_{1}={k}_{2}={J}_{2}$ with the now strict inequality ${J}_{2}<{J}_{1}$, while $|{k}_{3}|<{J}_{3}$. There are thus two options:(v.2.a) ${k}_{3}=-{J}_{2}$. Here the momentum condition reads $2{J}_{2}+(-{J}_{2})={J}_{1}+(-{J}_{3})$, which simplifies to ${J}_{2}=-{J}_{2}$, a contradiction.(v.2.b) ${k}_{3}=-{J}_{1}$. Here the momentum condition reads $2{J}_{2}+(-{J}_{1})={J}_{1}+(-{J}_{3})$, which simplifies to ${J}_{1}=3{J}_{2}$, thus ${J}_{3}=4{J}_{2}$. We turn to the resonance condition to find $2\sqrt{{J}_{2}}+\sqrt{3{J}_{2}}=\sqrt{3{J}_{2}}+\sqrt{4{J}_{2}}$, which is satisfied. In summary this leads to a 5-wave resonance parameterised by $K\in {\mathbb{Z}}_{+}$ as follows:$$K+K+(-3K)=3K+(-4K),\phantom{\rule{2.em}{0ex}}\sqrt{K}+\sqrt{K}+\sqrt{3K}=\sqrt{3K}+\sqrt{4K}\phantom{\rule{0.166667em}{0ex}}.$$

## Appendix B. Probe Calibration and Wavemaker Tests

**Figure A1.**Picture of the experimental tank with a close up on the installed probes. The wire of each capacitance probe is directly facing each wedge.

**Figure A3.**Tests of monochromatic wave generation by each wavemaker with the corresponding measurements by the closest probes. In each panel, the solid black line (red dashed line) represents the measurement by probe 1 (probe 2) of a wave generated by a monochromatic oscillation of the north wedge (south wedge). (

**a**) Monochromatic frequency $f=1.33$ Hz. (

**b**) Monochromatic frequency $f=2.08$ Hz. (

**c**) Monochromatic frequency $f=2.22$ Hz. (

**d**) Monochromatic frequency $f=2.30$ Hz.

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**Figure 1.**Poincaré cuts for the homogeneous version of the system consisting of two interacting quintets made out of three central modes and their negative versions.

**Figure 2.**The maximum efficiency $max{\mathcal{E}}_{{K}_{1}}$ as a function of starting ${K}_{3}$ (or ${f}_{3}$) for the numerical experiments considered. This demonstrates how narrow the peak transfer is at resonance in the discrete numerical case. (

**a**) We show $\beta =1$ (blue circles), $\beta =0.75$ (orange squares), and the encountering wave packets (‘A’ red triangle, defined in Equation (24), and ‘B’ green diamonds, defined in Equations (25)–(28)). Note the upper axis shows the equivalent frequency ${f}_{3}=\sqrt{g{K}_{3}}/2\pi $ and the (

**b**) plot shows the encountering wave packet case in isolation to better visualise the change in behaviour at resonance.

**Figure 3.**Time series for the efficiency ${\mathcal{E}}_{k}$ (23) (relative amplitude) for the ${K}_{1}$ target mode (

**a**) and ${K}_{3}$ (

**b**) under plane-wave propagation. A clear increase of energy transfer is observed in ${K}_{1}$ at the resonant frequency (${f}_{3}/\mathrm{Hz}\approx 2.206,$${K}_{3}=1248\frac{2\pi}{L}$) compared to off resonance (${f}_{3}/\mathrm{Hz}=2.25,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{K}_{3}=1298\frac{2\pi}{L}$ and ${f}_{3}/\mathrm{Hz}=2.15,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{K}_{3}=1185\frac{2\pi}{L}$).

**Figure 4.**Time dependent spatial Fourier spectra, $|{a}_{K}\left(t\right)|$, with the pertinent modes annotated, as in Figure 3 an increase of energy transfer is observed in ${K}_{1}$ at the resonant frequency (${f}_{3}/\mathrm{Hz}\approx 2.206,$${K}_{3}=1248\frac{2\pi}{L}$) compared to off resonance (${f}_{3}/\mathrm{Hz}=2.25,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{K}_{3}=1298\frac{2\pi}{L}$ and ${f}_{3}/\mathrm{Hz}=2.15,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{K}_{3}=1185\frac{2\pi}{L}$). The decrease in ${K}_{3}$ can be attributed to side-band instability, present in all cases, but not impeding the five-wave resonance.

**Figure 5.**Scaleograms for the plane-wave numerical probe data with $t=200$ s, produced by sampling 10,000 time points and performing a wavelet transform using Gabor wavelets with 48 oscillations, with a scale resolution of 11 frequency scales along with 64 voices per scale. Vertical lines represent the initial and final times t = 10 s, 190 s for the calculation of the efficiency to the target mode ${f}_{1}$ that avoids spurious boundary effects. Horizontal lines represent the theoretical frequencies ${f}_{1},{f}_{3}$ and ${f}_{4}$ stemming from the dispersion relation $f\left(k\right)=\sqrt{g\left|k\right|}/2\pi $ for the wavevectors ${K}_{1}={K}_{3}-{K}_{2}$, ${K}_{3}$ and ${K}_{4}={K}_{3}+{K}_{2}$. The width about the frequencies ${f}_{1}$ and ${f}_{4}$ is of the order $\Delta f=0.05f$.

**Figure 6.**Efficiency as a function of the experiment, parameterised by the corresponding wavevector ${K}_{3}$, for the plane-wave case for a long time series [0, 200 s]. The efficiency is defined as the sum of the squares of the scaleogram signals in Figure 5, on a strip of width $0.05{f}_{1}$ about ${f}_{1}$ and between the vertical lines in the figure, divided by the sum of squares over the whole range of frequencies, including ${f}_{2}$ which is not shown. An efficiency of $0.006$ corresponds to $0.6\%$ efficiency. The dashed grey vertical line corresponds to the theoretical resonant case ${K}_{1}:{K}_{2}:{K}_{3}=16:9:25$.

**Figure 7.**Joint probability density functions of the quintet phase ${\varphi}_{+}\left(t\right)$ versus its time derivative $\frac{d{\varphi}_{+}}{dt}\left(t\right)$, over the simulation time $t\in $ [0, 200 s], for the cases ${f}_{3}/\mathrm{Hz}=2.150,2.206$ and $2.250$ in the plane-wave case, corresponding to Figure 3 and Figure 5.

**Figure 8.**Snapshots of the surface elevation $\zeta (x,t)$ at various times the ‘A’ configuration (

**left**) and the ‘B’ configuration (

**right**) of initial coincident wave packets given in Equations (24)–(28). The numerical ‘probes’ are located at $x=198$ and $x=202$ to be 2 m either side of the domain centre, in-keeping with the experiments of Section 4.

**Figure 9.**Time series for the efficiency ${\mathcal{E}}_{k}$ (23) (relative amplitude) for the ${K}_{1}$ target mode for encountering wave-packets, case A on the left and B on the right. In case A the peak of energy transfer observed in ${K}_{1}$ is at the frequency (${f}_{3}/\mathrm{Hz}\approx 2.204,$${K}_{3}=1246\frac{2\pi}{L}$) and is here compared to off resonance (${f}_{3}/\mathrm{Hz}=2.22,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{K}_{3}=1264\frac{2\pi}{L}$ and ${f}_{3}/\mathrm{Hz}=2.185,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{K}_{3}=1224\frac{2\pi}{L}$). In case B the peak of energy transfer observed in ${K}_{1}$ is at the frequency (${f}_{3}/\mathrm{Hz}\approx 2.22,$${K}_{3}=1264\frac{2\pi}{L}$) and is here compared to off resonance (${f}_{3}/\mathrm{Hz}=2.25,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{K}_{3}=1298\frac{2\pi}{L}$ and ${f}_{3}/\mathrm{Hz}=2.172,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{K}_{3}=1210\frac{2\pi}{L}$).

**Figure 10.**Scaleograms for measurements of probe 1 (

**left panels**) and probe 2 (

**right panels**), for three numerical wave packet experiments close to the resonance. Frequencies are displayed in the vertical axis and time in the horizontal axis. Horizontal lines represent the theoretical frequencies ${f}_{1},{f}_{3}$ and ${f}_{4}$ stemming from the dispersion relation $f\left(k\right)=\sqrt{g\left|k\right|}/2\pi $ for the wavevectors ${K}_{1}={K}_{3}-{K}_{2}$, ${K}_{3}$ and ${K}_{4}={K}_{3}+{K}_{2}$. The rows from top to bottom correspond to experiments with ${f}_{3}/\mathrm{Hz}=2.185,2.204,2.220$ chosen to correspond to the time series in Figure 9. Vertical lines represent the initial and final times t = 10 s, 190 s for the calculation of the efficiency to the target mode ${f}_{1}$ that avoids spurious boundary effects.

**Figure 11.**Efficiency as a function of the numerical experiment, for each probe, parameterised by the corresponding wavevector ${K}_{3}$, for the encountering wavepackets case for a long time series [0, 200 s]. The efficiency is defined as the sum of the squares of the scaleogram signals in Figure 10, on a strip of width $0.05{f}_{1}$ about ${f}_{1}$ and between the vertical lines in the figure, divided by the sum of squares over the whole range of frequencies, including ${f}_{2}$ which is not shown. An efficiency of $0.10$ corresponds to $10\%$ efficiency. The dashed grey vertical line corresponds to the theoretical resonant case ${K}_{1}:{K}_{2}:{K}_{3}=16:9:25$.

**Figure 12.**Diagram of the experimental setup, including probe #1 and probe #2. Note N denotes north end; S indicates south end. Unit: Metres.

**Figure 13.**Plots of the probe measurements for experiment number 7. The vertical lines correspond to the first time ${t}_{0}$ the wavepackets with wavevectors $\pm {K}_{3}$ arrive at the probes (leftmost line), and the time ${t}_{1}$ the wavepackets with wavevectors $\pm {K}_{2}$ arrive at the probes after reflection from the walls of the tank (rightmost line).

**Figure 14.**Scaleograms for the artificial signal, Equation (29), mimicking three of the main frequencies found in the experiments: ${f}_{1},{f}_{3},{f}_{4}$. The width about the frequencies ${f}_{1}$ and ${f}_{4}$ is of the order $\Delta f=0.09f$ and is due to the way the scaleogram is produced.

**The following comments apply to all scaleogram figures in this section.**The colormaps indicate the amplitude of the scaleograms. The scaleograms are produced by sampling the artificial signal using 4096 time points and performing a continuous wavelet transform using Gabor wavelets with 96 oscillations, with a scale resolution of 11 frequency scales along with 256 voices per scale.

**Figure 15.**Scaleograms for measurements of probe 1 (

**left panels**) and probe 2 (

**right panels**), for experiments 1, 2 and 3. Frequencies are displayed in the vertical axis and time in the horizontal axis. Three of the main frequencies found in the experiments are marked in each panel: ${f}_{1},{f}_{3},{f}_{4}$, along with ${f}_{1}+{f}_{2}$ for reference. The rows from top to bottom correspond to experiments 1, 2 and 3: Starting from the top, ${f}_{3}/\mathrm{Hz}=2.05,2.08,2.1$. The three vertical lines correspond, from left to right, to: (i) The theoretical time (which depends on the experiment) when the first nonlinearly-produced wavepackets with central wavenumbers $\pm {K}_{1}$ arrive at the probes; (ii) the theoretical time (which is fixed) when the first wavepackets with central wavenumbers $\pm {K}_{2}$ arrive at the probes after reflecting from the ends of the tank; (iii) the fixed time when the artificial signal’s scalogram shows a significant departure from the expected frequencies. See further comments on the colormaps and details of scaleogram calculations in the caption of Figure 14.

**Figure 16.**Same description as in the caption of Figure 15, but for experiments 4, 5 and 6, corresponding to ${f}_{3}/\mathrm{Hz}=2.15,2.185,2.22$, respectively.

**Figure 17.**Same description as in the caption of Figure 15, but for experiments 7, 8 and 9, corresponding to ${f}_{3}/\mathrm{Hz}=2.25,2.3,2.35$, respectively.

**Figure 18.**Efficiency as a function of the experiment, for the two probes and for early (dashed lines) and late (solid lines) measurements. An efficiency of $0.005$ corresponds to $0.5\%$ efficiency. The initial time for the early measurements is the theoretical time (which depends on the experiment) when the first nonlinearly-produced wavepackets with central wavenumbers $\pm {K}_{1}$ arrive at the probes: For the 9 experiments, ${t}_{0}=\{14.44,15.04,15.44,16.44,17.14,17.84,18.42,19.42,20.42\}$ s. The final time for the early measurements is the theoretical time (which is fixed) when the first wavepackets with central wavenumbers $\pm {K}_{2}$ arrive at the probes after reflecting from the ends of the tank. The dashed grey vertical line corresponds to the theoretical resonant case ${K}_{1}:{K}_{2}:{K}_{3}=16:9:25$.

**Figure 19.**Phase-locking values of analytic-signal-approach triad phases, Equations (30) and (31), obtained from probe data, for different experiments within the three scenarios studied in this paper: (

**a**) Encountering numerical plane waves; (

**b**) encountering numerical wave packets; (

**c**) encountering experimental waves. In all three scenarios, a clear local maximum of the phase-locking value is attained at the predicted resonance, corresponding to the maximum in efficiency for each case, cf. Figure 6, Figure 11 and Figure 18, respectively. In all panels, the dashed grey vertical line corresponds to the theoretical resonant case ${K}_{1}:{K}_{2}:{K}_{3}=16:9:25$.

**Table 1.**For each experiment, frequency ${f}_{3}$ along with the wavemaker oscillation amplitude ${A}_{3}$, chosen so that ${A}_{3}{K}_{3}\approx 0.12$, where ${K}_{3}=4{\pi}^{2}{g}^{-1}{f}_{3}^{2}$ is the wavenumber. In the last column, the theoretical target frequency ${f}_{1}=\sqrt{g{K}_{1}}/2\pi $ that would be observed if wavenumber ${K}_{1}={K}_{3}-{K}_{2}$ was produced. We highlight in boldface the resonant case ${f}_{1}:{f}_{2}:{f}_{3}=4:3:5$, corresponding to Equation (16).

Experiment | ${\mathit{f}}_{3}$ (Hz) | ${\mathit{A}}_{3}$ (cm) | Target ${\mathit{f}}_{1}$ (Hz) |
---|---|---|---|

Exp 1 | 2.05 | 0.71 | 1.56 |

Exp 2 | 2.08 | 0.69 | 1.60 |

Exp 3 | 2.10 | 0.68 | 1.62 |

Exp 4 | 2.15 | 0.64 | 1.69 |

Exp 5 | 2.185 | 0.62 | 1.73 |

Exp 6 | 2.22 | 0.60 | 1.78 |

Exp 7 | 2.25 | 0.59 | 1.81 |

Exp 8 | 2.30 | 0.56 | 1.88 |

Exp 9 | 2.35 | 0.54 | 1.94 |

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**MDPI and ACS Style**

Lucas, D.; Perlin, M.; Liu, D.-Y.; Walsh, S.; Ivanov, R.; Bustamante, M.D.
Five-Wave Resonances in Deep Water Gravity Waves: Integrability, Numerical Simulations and Experiments. *Fluids* **2021**, *6*, 205.
https://doi.org/10.3390/fluids6060205

**AMA Style**

Lucas D, Perlin M, Liu D-Y, Walsh S, Ivanov R, Bustamante MD.
Five-Wave Resonances in Deep Water Gravity Waves: Integrability, Numerical Simulations and Experiments. *Fluids*. 2021; 6(6):205.
https://doi.org/10.3390/fluids6060205

**Chicago/Turabian Style**

Lucas, Dan, Marc Perlin, Dian-Yong Liu, Shane Walsh, Rossen Ivanov, and Miguel D. Bustamante.
2021. "Five-Wave Resonances in Deep Water Gravity Waves: Integrability, Numerical Simulations and Experiments" *Fluids* 6, no. 6: 205.
https://doi.org/10.3390/fluids6060205