# Coupling Conditions for Water Waves at Forks

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. General Scalar Nonlinear Wave Equations

## 3. The Nonlinear Shallow Water Equations

#### 3.1. Conserved Quantities

#### 3.2. Small Amplitude Limit

## 4. Reduction of the Shallow Water Equations

#### 4.1. Mass Flux

#### 4.2. Energy Flux

#### 4.3. Momentum Flux for a General Fork

#### 4.4. Momentum Flux for the T-Fork

#### 4.5. Effective 1D Model for the T-Fork

## 5. Numerical Solutions of the 2D Shallow Water Equations

#### 5.1. Wave Incident into Branch 1

#### 5.1.1. Small Amplitude Waves $a/d=0.1$

#### 5.1.2. Very Large Amplitude Waves $a/d=2$

#### 5.2. Wave Incident into Branch 3

#### 5.2.1. Small Amplitude Waves $a/d=0.1$

#### 5.2.2. Large Amplitude Waves $a/d=1$

## 6. Discussion and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Time evolution of the energies ${E}_{2}^{i}$ for the kink motion in branches $i=1,2$ for the T-junction (90 degrees) in full line (red online), for the Y-junction (45 degrees) in dashed line. The energy ${E}_{1}^{i}$ for the 1D effective model is plotted with points.

**Figure 4.**Time evolution of the wave mass ${M}_{w}$ (

**left**) and the wave energy ${E}_{w}$ (

**right**) for a wave incident in branch 1 for $a/d=0.1$.

**Figure 5.**Time evolution of the mass and energy quantities $\delta m$ (black online) and $\phantom{\rule{3.33333pt}{0ex}}\delta e$ (red online) for $a/d=0.1$.

**Figure 7.**Snapshot of the surface elevation h at time $t=0.9$ for a wave incident in branch 1 for $a/d=2$.

**Figure 8.**Time evolution of the wave mass ${M}_{w}$ (

**left**) and the wave energy ${E}_{w}$ (

**right**) for a wave incident in branch 1 for $a/d=2$.

**Figure 9.**Time evolution of the mass and energy quantities $\delta m,\phantom{\rule{3.33333pt}{0ex}}\delta e$ for $a/d=2$.

**Figure 11.**Time evolution of the wave mass ${M}_{w}$ (

**left**) and the wave energy ${E}_{w}$ (

**right**) for a wave incident in branch 3 for $a/d=0.1$ (

**top panels**) and $a/d=2$ (

**bottom panels**). Notice the different scales.

**Figure 12.**Time evolution of the mass and energy quantities $\delta m$ (black online) and $\phantom{\rule{3.33333pt}{0ex}}\delta e$ (red online) for $a/d=0.1$.

**Figure 13.**Time evolution of ${h}_{1},{h}_{2}$ and ${h}_{3}$ (top) and ${u}_{3},{u}_{2},{u}_{1},{v}_{1}$ (bottom) for $a/d=0.1$.

**Figure 14.**Snapshot of the surface elevation h at time $t=0.8$ for a wave incident in branch 3 for $a/d=2$.

**Figure 15.**Time evolution of the mass and energy quantities $\delta m$ (black online) and $\delta e$ (purple online) for $a/d=2$.

**Figure 16.**Time evolution of the x and y momenta quantities $\delta {p}_{x}$ (black online) and $\delta {p}_{y}$ (purple online) for $a/d=1$.

**Figure 17.**Time evolution of ${h}_{1},{h}_{2},{h}_{3}$ (

**top**) and ${u}_{3},{u}_{2},{u}_{1},{v}_{1}$ (

**bottom**) for $a/d=1$.

**Figure 19.**Time evolution of the quantity ${v}_{1m}$ obtained from the mass conservation law (49) (purple online) and ${v}_{1}$ from the 2D numerical solution (black online).

Type | Known | Unknown |
---|---|---|

wave in branch 1 | ${h}_{1},{v}_{1}$ | ${h}_{2},{u}_{2},{h}_{3},{u}_{3}$ |

wave in branch 3 | ${h}_{3},{u}_{3}$ | ${h}_{1},{v}_{1},{h}_{2},{u}_{2}$ |

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**MDPI and ACS Style**

Caputo, J.; Dutykh, D.; Gleyse, B.
Coupling Conditions for Water Waves at Forks. *Symmetry* **2019**, *11*, 434.
https://doi.org/10.3390/sym11030434

**AMA Style**

Caputo J, Dutykh D, Gleyse B.
Coupling Conditions for Water Waves at Forks. *Symmetry*. 2019; 11(3):434.
https://doi.org/10.3390/sym11030434

**Chicago/Turabian Style**

Caputo, Jean–Guy, Denys Dutykh, and Bernard Gleyse.
2019. "Coupling Conditions for Water Waves at Forks" *Symmetry* 11, no. 3: 434.
https://doi.org/10.3390/sym11030434