# Note on the Application of Transient Wave Packets for Wave–Ice Interaction Experiments

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## Abstract

**:**

## 1. Introduction

## 2. Theoretical Basis

#### 2.1. Open Water Waves

#### 2.2. Ice-Covered Waves

#### 2.3. Transient Wave Packets

#### 2.4. Wave Damping

## 3. Experiment

#### 3.1. Model Ice

#### 3.2. Experimental Setup

#### 3.3. Experimental Programme

## 4. Results

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

**Figure A1.**Overview on the analysis procedure and measurement results of the investigated regular waves. (

**a**) Example analysis procedure for wave Reg I. The top diagram displays the whole measurement as a black curve and the selected evaluation period as a blue curve. The bottom diagram presents the selected period in detail with the measurement result as a black curve, the selected evaluation period as a blue curve and the identified zero-upcrossing periods for the detailed analysis as a red dashed curve. (

**b**) Example analysis procedure for wave Reg IV. The top diagram displays the whole measurement as a black curve and the selected evaluation period as a blue curve. The bottom diagram presents the selected period in detail with the measurement result as a black curve, the selected evaluation period as a blue curve and the identified zero-upcrossing periods for the detailed analysis as a red dashed curve. (

**c**) Results of the analysis—course of the wave heights (

**top left**), wave periods (

**top centre**) and wave crest asymmetries (

**top right**)—for the four regular waves at each measuring position for the identified evaluation periods. The respective variance of the statistical quantities is illustrated in the bottom diagrams.

## Appendix A

Ultralab USS 30250 | Ultralab USS 2001300 | |
---|---|---|

Blind area | $30\phantom{\rule{0.166667em}{0ex}}\mathrm{mm}$ | $200\phantom{\rule{0.166667em}{0ex}}\mathrm{mm}$ |

Working range | $250\phantom{\rule{0.166667em}{0ex}}\mathrm{mm}$ | $1300\phantom{\rule{0.166667em}{0ex}}\mathrm{mm}$ |

Techn. resolution | $0.18\phantom{\rule{0.166667em}{0ex}}\mathrm{mm}$ | $0.36\phantom{\rule{0.166667em}{0ex}}\mathrm{mm}$ |

Reproducibility | $\pm \text{0.15\%}$ | $\pm 1\phantom{\rule{0.166667em}{0ex}}\mathrm{mm}$ |

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**Figure 1.**Definition of the coordinate system showing the profile of the x-z plane: The point of origin is at the still water level, the positive z-axis faces upwards, opposing the gravitational force, and the positive x-axis points to the direction of wave propagation assuming uni-directional waves. Note that the wave elevation in open water and the vertical displacement of the ice are assigned the same variable $\zeta $, but as soon as a clear distinction has to be made between open water and ice-covered water, this is indicated by the indices i for ice and w for open water.

**Figure 2.**Overview of the TWP concept. (

**a**) Theoretical TWP concept—normalised TWP Fourier spectrum (

**left**) and normalised TWP in time domain at concentration point (

**right**). (

**b**) Application TWP concept—TWP at concentration point (

**top right**) and corresponding surface elevation at the wave board (

**bottom right**). The surface elevation based on open water dispersion is shown by the blue curve and the consequences for wave propagation due to the presence of the ice sheet are illustrated by the red curve. The comparison of the dispersion relation between open water (blue curve) and the ice sheet (red curve) is shown in the left diagram.

**Figure 4.**Experimental results of the TWP. The diagrams on the left-hand side present the measured vertical displacement of the ice for specific locations along the tank. The centre diagrams show the corresponding amplitude spectrum and the diagrams on the right-hand side show the determined attenuation coefficient. The four coloured vertical lines in the diagrams on the right-hand side represent the position of the investigated regular waves within the TWP frequency spectrum.

**Figure 5.**Presentation of the experimentally determined attenuation and damping coefficients. The attenuation coefficient for four selected frequencies is shown in (

**a**), comparing the results of regular waves with the TWP measurements of the same frequency. The exponential damping coefficient determined by the TWP technique is presented in (

**b**). (

**a**) Comparison of the attenuation coefficients for selected frequencies of the TPW (plus) and the corresponding four investigated regular waves (stars). (

**b**) Frequency-dependent exponential damping coefficient a(wn).

**Figure 6.**Comparison between measured vertical displacement of the ice and analytical solution of the linear theory for selected locations. The results in the time domain are shown on the left-hand side and the corresponding amplitude spectra are compared on the right hand side. For linear theory, results for the classical approach as well as for the extended approach by taking the measured damping into account are shown.

Parameter | Test Conditions |
---|---|

Water density | ${\rho}_{w}=1005.5\phantom{\rule{0.166667em}{0ex}}{\mathrm{Kgm}}^{-3}$ |

Ice density | ${\rho}_{i}=846\phantom{\rule{0.166667em}{0ex}}{\mathrm{Kgm}}^{-3}$ |

Ice thickness | $h=0.0255\phantom{\rule{0.166667em}{0ex}}\mathrm{m}$ |

Ice sheet length | $\Delta {x}_{i}=51\phantom{\rule{0.166667em}{0ex}}\mathrm{m}$ |

Bending strength | ${\sigma}_{B}=36.3$–$37.5\phantom{\rule{0.166667em}{0ex}}\mathrm{kPa}$ |

Effective strain modulus | $E=2.54$–$3.08\phantom{\rule{0.166667em}{0ex}}\mathrm{MPa}$ |

**Table 2.**Overview on the investigated four regular waves. The first table column presents the unique identifier for each regular wave, the second column the angular frequency and the last column the wave steepness.

Wave ID | $\mathit{\omega}\phantom{\rule{0.166667em}{0ex}}[\mathbf{rad}\phantom{\rule{-1.111pt}{0ex}}/\phantom{\rule{-0.55542pt}{0ex}}\mathbf{s}]$ | ${\mathit{\u03f5}}_{\mathbf{w}}\phantom{\rule{0.166667em}{0ex}}\left[\mathbf{m}\right]$ |
---|---|---|

Reg I | $4.29$ | $0.025$ |

Reg II | $4.97$ | $0.025$ |

Reg III | $5.63$ | $0.025$ |

Reg IV | $6.33$ | $0.025$ |

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**MDPI and ACS Style**

Klein, M.; Hartmann, M.; von Bock und Polach, F.
Note on the Application of Transient Wave Packets for Wave–Ice Interaction Experiments. *Water* **2021**, *13*, 1699.
https://doi.org/10.3390/w13121699

**AMA Style**

Klein M, Hartmann M, von Bock und Polach F.
Note on the Application of Transient Wave Packets for Wave–Ice Interaction Experiments. *Water*. 2021; 13(12):1699.
https://doi.org/10.3390/w13121699

**Chicago/Turabian Style**

Klein, Marco, Moritz Hartmann, and Franz von Bock und Polach.
2021. "Note on the Application of Transient Wave Packets for Wave–Ice Interaction Experiments" *Water* 13, no. 12: 1699.
https://doi.org/10.3390/w13121699