# Parasitic Capillary Waves on Small-Amplitude Gravity Waves with a Linear Shear Current

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## Abstract

**:**

## 1. Introduction

## 2. Formulation of the Problem

#### 2.1. Formulation in the Physical Plane

#### 2.2. Unsteady Conformal Mapping of the Flow Domain

## 3. The Method of Computation

## 4. Numerical Results

#### 4.1. Initial Values

#### 4.2. Generation of Parasitic Capillary Waves

**a**) $\gamma =0.2$ and $\Omega =-4.0$ with $\alpha =\mathrm{O}\left({10}^{-3}\right)$ and(

**b**) $\gamma =0.1$ and $\Omega =-8.0$ with $\alpha =\mathrm{O}\left({10}^{-4}\right)$ where the values of the wave steepness $\alpha $ are summarized in Table 1.

**a**) $\gamma =0.2$ and $\Omega =-4.0$ and (

**b**) $\gamma =0.1$ and $\Omega =-8.0$, the wave energy is transferred from gravity waves to capillary waves. These results indicate that, even for very small-amplitude waves such as the wave steepness $\alpha =\mathrm{O}\left({10}^{-3}\right)$ or $\mathrm{O}\left({10}^{-3}\right)$, a nonlinearity works for generation of parasitic capillary waves.

## 5. Discussions

**a**) $\gamma =0.2$ and $\Omega =-4.0$ and (

**b**) $\gamma =0.1$ and $\Omega =-8.0$. It is found that the weakly nonlinear solutions using (31) and (32) reasonably well approximate the fully nonlinear solutions using (12) and (13) for small-amplitude waves. Therefore, the approximate evolution equations given by (31) and (32) would be useful to study the generation of parasitic capillary waves on small-amplitude gravity waves with a linear shear current.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Two-dimensional periodic motion of deep-water gravity-capillary waves on a linear shear current and conformal mapping of the flow domain into a complex plane. (

**a**) The z-plane ($z=x+\mathrm{i}y$); (

**b**) The $\zeta $-plane ($\zeta =\xi +\mathrm{i}\eta $).

**Figure 2.**Pure gravity steady waves progressing in permanent form on a linear shear current with the shear strength $\Omega $. The slope ${\theta}^{\left(0\right)}\left(x\right)$ and the curvature ${\kappa}^{\left(0\right)}\left(x\right)$ are given by (26), and the parameter $\gamma $ is defined by (24). The number of Fourier modes in (25) is set to $M=128$. (

**a**) $\gamma =0.2$ (black line — : $\Omega =0.0$, red line — : $\Omega =-1.0$, blue line — : $\Omega =-2.0$, green line — : $\Omega =-3.0$, magenta line — : $\Omega =-4.0$). (

**b**) $\gamma =0.1$ (black line — : $\Omega =0.0$, red line — : $\Omega =-2.0$, blue line — : $\Omega =-4.0$, green line — : $\Omega =-6.0$, magenta line — : $\Omega =-8.0$).

**Figure 3.**Time evolution of the wave profile $\tilde{y}(x,t)$ and the slope $\theta (x,t)$ of gravity-capillary waves on a linear shear current with the shear strength $\Omega $. The initial values are set to pure gravity waves in Figure 2. See Table 1 for the parameter values of the initial waves. The parameter $\gamma $ is defined by (24). $1/{W}_{e}=0.006$, $N=256$ and $\Delta t=2\pi /2048$. (

**a**) $\gamma =0.2$ (red line — : $\Omega =-1.0$, blue line — : $\Omega =-2.0$, green line — : $\Omega =-3.0$, magenta line — : $\Omega =-4.0$). (

**b**) $\gamma =0.1$ (red line — : $\Omega =-2.0$, blue line — : $\Omega =-4.0$, green line — : $\Omega =-6.0$, magenta line — : $\Omega =-8.0$).

**Figure 4.**Time evolution of the surface energy ratio ${E}_{\mathrm{S}}\left(t\right)/{E}_{\mathrm{T}}\left(t\right)$ of the computed results in Figure 1, Figure 2, Figure 3. The surface energy ${E}_{\mathrm{S}}\left(t\right)$ and the total energy ${E}_{\mathrm{T}}\left(t\right)$ are given by (18) and (23), respectively. The parameter $\gamma $ is defined by (24). (

**a**) $\gamma =0.2$ (red line — : $\Omega =-1.0$, blue line — : $\Omega =-2.0$, green line — : $\Omega =-3.0$, magenta line — : $\Omega =-4.0$). (

**b**) $\gamma =0.1$ (red line — : $\Omega =-2.0$, blue line — : $\Omega =-4.0$, green line — : $\Omega =-6.0$, magenta line — : $\Omega =-8.0$).

**Figure 5.**Time evolution of the energy spectrum ${S}_{k}\left(t\right)$ of the computed results in Figure 3. ${S}_{k}\left(t\right)$ is given by (27) and the parameter $\gamma $ for is defined by (24). (black line — : $t=0$, red line — : $t=\pi $, blue line — : $t=2\pi $, green line — : $t=3\pi $). (

**a**) $\gamma =0.2$; (

**b**) $\gamma =0.1$.

**Figure 6.**Comparison of the computed results of the wave profile $\tilde{y}(x,t)$ and the slope $\theta (x,t)$ at $t=3\pi $ using the approximate free surface conditions (31) and (32) with those using the fully nonlinear conditions (12) and (13). The initial values are set to pure gravity waves in Figure 2. See Table 1 for the parameter values of the initial waves. The parameter $\gamma $ is defined by (24). $1/{W}_{e}=0.006$, $N=256$ and $\Delta t=2\pi /2048$. (black line — : the fully nonlinear solutions using (12) and (13), red line — : the weakly nonlinear solutions using (31) and (32)). (

**a**) $\gamma =0.2$ and $\Omega =-4.0$; (

**b**) $\gamma =0.1$ and $\Omega =-8.0$.

$\mathit{\gamma}$ | $\mathsf{\Omega}$ | c | $\mathit{\alpha}$ |
---|---|---|---|

0.2 | 0.0 | 1.00604 × 10${}^{+0}$ | 3.49321 × 10${}^{-2}$ |

0.2 | −1.0 | 6.22553 × 10${}^{-1}$ | 1.32682 × 10${}^{-2}$ |

0.2 | −2.0 | 4.19196 × 10${}^{-1}$ | 5.90622 × 10${}^{-3}$ |

0.2 | −3.0 | 3.08239 × 10${}^{-1}$ | 3.11534 × 10${}^{-3}$ |

0.2 | −4.0 | 2.41766 × 10${}^{-1}$ | 1.86715 × 10${}^{-3}$ |

0.1 | 0.0 | 1.00138 × 10${}^{+0}$ | 1.67068 × 10${}^{-2}$ |

0.1 | −2.0 | 4.15398 × 10${}^{-1}$ | 2.85668 × 10${}^{-3}$ |

0.1 | −4.0 | 2.37640 × 10${}^{-1}$ | 9.20809 × 10${}^{-4}$ |

0.1 | −6.0 | 1.64103 × 10${}^{-1}$ | 4.30888 × 10${}^{-4}$ |

0.1 | −8.0 | 1.24961 × 10${}^{-1}$ | 2.45879 × 10${}^{-4}$ |

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**MDPI and ACS Style**

Murashige, S.; Choi, W.
Parasitic Capillary Waves on Small-Amplitude Gravity Waves with a Linear Shear Current. *J. Mar. Sci. Eng.* **2021**, *9*, 1217.
https://doi.org/10.3390/jmse9111217

**AMA Style**

Murashige S, Choi W.
Parasitic Capillary Waves on Small-Amplitude Gravity Waves with a Linear Shear Current. *Journal of Marine Science and Engineering*. 2021; 9(11):1217.
https://doi.org/10.3390/jmse9111217

**Chicago/Turabian Style**

Murashige, Sunao, and Wooyoung Choi.
2021. "Parasitic Capillary Waves on Small-Amplitude Gravity Waves with a Linear Shear Current" *Journal of Marine Science and Engineering* 9, no. 11: 1217.
https://doi.org/10.3390/jmse9111217