# An Experimental and Numerical Study of Long Wave Run-Up on a Plane Beach

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## Abstract

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## 1. Introduction

## 2. Methodology

**Figure 1.**Schematic cross section of the considered setup: A constant depth region is attached to a linearly sloping beach with angle $\gamma =tan\alpha $, where γ is the beach slope. Also indicated are the wave gauges (WG) from the experimental setup.

#### 2.1. Theoretical Background

#### 2.2. Physical Model

**Figure 2.**Schematic drawing of the wave flume and its components with position indication of instrumentation. Adapted from [38], with permission from © 2012 World Scientific Publishing Company Incorporated.

**Figure 3.**Stages of the image processing with raw data images and final result after image processing routines. (

**a**) First camera scene; (

**b**) Second camera scene; (

**c**) Final stitched image for analysis.

#### 2.3. Scale Effects

#### 2.4. Numerical Model

#### 2.5. Boundary Conditions

**Table 1.**Characteristics of sinusoidal waves used, surf similarity according to Equation (5), naming scheme indicates period and wave height, respectively.

Wave-ID | Height | Period | Length | Non-linearity | Rel. Amplitude | Surf Similarity |
---|---|---|---|---|---|---|

$(-)$ | $H\left(m\right)$ | $T\left(s\right)$ | $L\left(m\right)$ | $\u03f5=\frac{{A}_{0}}{h}(-)$ | $\frac{{A}_{0}}{L}(-)$ | $\xi (-)$ |

T20_H2 | 0.02 | 20.00 | 34.31 | 0.033 | 2.91 × 10${}^{-4}$ | 4.42 |

T30_H2 | 0.02 | 30.00 | 51.47 | 0.033 | 1.94 × 10${}^{-4}$ | 6.63 |

T44_H3 | 0.03 | 44.00 | 75.48 | 0.050 | 1.99 × 10${}^{-4}$ | 7.94 |

T58_H3 | 0.03 | 58.00 | 99.50 | 0.050 | 1.51 × 10${}^{-4}$ | 10.46 |

T77_H4 | 0.04 | 77.00 | 132.09 | 0.067 | 1.51 × 10${}^{-4}$ | 12.03 |

T100_H4 | 0.04 | 100.00 | 171.55 | 0.067 | 1.17 × 10${}^{-4}$ | 15.62 |

## 3. Results

#### 3.1. Quality of Experimentally Generated Waves

**Figure 4.**Time series of surface elevation for the reference curve (red dashed) and measured values (solid lines) for all six runs of T20_H2 (

**top left**); T30_H2 (

**top right**); T58_H3 (

**bottom left**) and T100_H4 (

**bottom right**) as measured by the pressure sensor at the water inlet.

**Table 2.**Variation of Brier score [52] and standard deviation (STD) of all generated waves.

Wave-ID | Brier Score | STD (cm) |
---|---|---|

T20_H2 | 0.65–0.71 | 0.15–0.19 |

T30_H2 | 0.30–0.38 | 0.10–0.11 |

T44_H3 | 0.30–0.40 | 0.18–0.20 |

T58_H3 | 0.22–0.28 | 0.15–0.17 |

T77_H4 | 0.35–0.45 | 0.24–0.29 |

T100_H4 | 0.29–0.48 | 0.24–0.34 |

**Figure 5.**Time series of surface elevation η at wave gauges WG1 (left), WG2 (middle) and WG3 (right) for four different wave shapes. From top to bottom: T20_H2, T30_H2, T58_H3, T100_H4. Experimental data (blue), numerical results with experimental boundary data (black dash-dotted), numerical results with analytical boundary data shifted by ${t}_{\mathrm{shift}}=(4.2/\sqrt{g{h}_{0}}+10)\text{s}$ to fit the experimental data (red dashed).

#### 3.2. Validation of the Numerical Model

**Figure 6.**Time series of run-up elevation R (top) and velocity V (bottom) using analytical boundary data for three different wave shapes (

**left**: T20_H2;

**middle**: T44_H3;

**right**: T100_H4. Depicted are the theoretical evolution for periodic sinusoidal waves (red dashed), the numerical simulation with a single cycle sinusoidal wave (black dash-dotted) and the simulation with periodic sinusoidal waves (blue solid). Also shown are the positions of the theoretical maximum/minimum run-up elevations (red circles) and velocities (red diamonds) in the first period. The shaded regions depict the intervals where the extreme values for the diagrams in Figure 9 and Figure 10 were computed.

#### 3.3. Shoreline Motion

**Figure 7.**Time series of run-up elevation R from the experiments and the numerical simulation using experimental boundary data for all six wave shapes (left to right: T20_H2, T30_H2, T44_H3

**(top row**); T58_H3

**(bottom left**); T77_H4, T100_H4 (

**bottom row**)). Depicted are the theoretical evolution for periodic sinusoidal waves (red dashed), the numerical simulation (black dash-dotted) and the results from the six experimental measurements (solid lines). The shaded regions depict the intervals where the extreme values for the diagrams in Figure 9 were computed.

#### 3.3.1. Run-up and Run-down of Long Waves

#### 3.3.2. Shoreline Velocity during Run-up and Run-down

Wave-ID | Analyt. BC, Periodic Wave | Experiments/exp. BC, Single Cycle Wave | |||
---|---|---|---|---|---|

All Extreme Values | ${R}_{\mathrm{up}}$ | ${R}_{\mathrm{down}}$ | ${V}_{\mathrm{up}}$ | ${V}_{\mathrm{down}}$ | |

(s) | (s) | (s) | (s) | (s) | |

T20_H2 | 30–80 | 40–52 | – | 40–52 | – |

T30_H2 | 30–80 | 40–57 | – | 40–57 | – |

T44_H3 | 35–120 | 43–70 | – | 43–54 | 60–80 |

T58_H3 | 40–150 | 45–80 | – | 45–67 | – |

T77_H4 | 40–200 | 50–100 | – | 50–74 | – |

T100_H4 | 50–250 | 60–110 | – | – | – |

#### 3.4. Extremal Shoreline Dynamics

**Figure 9.**Maximum run-up (

**left**) and maximum run-down (

**right**) of long sinusoidal waves. Analytical (black stars), experimental (red circles) and numerical (diamonds)—with experimental (cyan) and analytical (red) boundary condition. Solid black line according to Equation (7). Colored solid lines are computed according to Equation (6).

**Figure 10.**Maximum shoreline velocities during run-up (

**left**) and run-down (

**right**). Analytical (black stars), experimental (red circles) and numerical (diamonds)—with experimental (cyan) and analytical (red) boundary condition. Solid black line according to Equation (7). Colored solid lines are computed according to Equation (6).

#### 3.4.1. Shoreline Location

#### 3.4.2. Shoreline Velocity

## 4. Discussion and Conclusions

## Supplementary Materials

Supplementary File 1## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix

- Time: experiment time in [s], range: 0 s to 120 s,
- Reference value: analytical surface elevation at the water inlet ($x=31.92\text{\hspace{0.17em}m}$) in [cm], ideal single cycle sinusoidal wave,
- Actual value: measured surface elevation at the water inlet ($x=31.92\text{\hspace{0.17em}m}$) in [cm] as generated by the wave generator,
- WG1, WG2, WG3: surface elevation in [cm] as measured at the three wave gauges,
- Velocity: measured wave velocity at WG1 in [cm/s] in propagation direction.

- t: time of measurement for shoreline position in [s]; duration depending on wave period,
- pos: horizontal position of shoreline at time t in [m] (not run-up height!),
- v: computed shoreline velocity in [cm/s].

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**MDPI and ACS Style**

Drähne, U.; Goseberg, N.; Vater, S.; Beisiegel, N.; Behrens, J.
An Experimental and Numerical Study of Long Wave Run-Up on a Plane Beach. *J. Mar. Sci. Eng.* **2016**, *4*, 1.
https://doi.org/10.3390/jmse4010001

**AMA Style**

Drähne U, Goseberg N, Vater S, Beisiegel N, Behrens J.
An Experimental and Numerical Study of Long Wave Run-Up on a Plane Beach. *Journal of Marine Science and Engineering*. 2016; 4(1):1.
https://doi.org/10.3390/jmse4010001

**Chicago/Turabian Style**

Drähne, Ulrike, Nils Goseberg, Stefan Vater, Nicole Beisiegel, and Jörn Behrens.
2016. "An Experimental and Numerical Study of Long Wave Run-Up on a Plane Beach" *Journal of Marine Science and Engineering* 4, no. 1: 1.
https://doi.org/10.3390/jmse4010001