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

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 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].

## References

- Wang, X.; Liu, P.L. An analysis of 2004 Sumatra earthquake fault plane mechanisms and Indian Ocean tsunami. J. Hydraul. Res.
**2006**, 44, 147–154. [Google Scholar] [CrossRef] - Geophysical Loss Events Worldwide 1980–2014; NatCatSERVICE Munich Re: Munich, Germany, 2015.
- Synolakis, C.E.; Kong, L. Runup measurements of the December 2004 Indian Ocean tsunami. Earthq. Spectra
**2006**, 22, 67–91. [Google Scholar] [CrossRef] - Synolakis, C.E.; Bernard, E.N. Tsunami science before and beyond Boxing Day 2004. Philos. Trans. R. Soc. Lond. A Math. Phys. Eng. Sci.
**2006**, 364, 2231–2265. [Google Scholar] [CrossRef] [PubMed] - Briggs, M.J.; Synolakis, C.E.; Hughes, S.A. Laboratory measurements of tsunami run-up, 1993. In Proceedings of the Tsunami, Wakayama, Japan, 23–27 August 1993.
- Gedik, N.; Irtem, E.; Kabdasli, S. Laboratory investigation of tsunami run-up. Ocean Eng.
**2005**, 32, 513–528. [Google Scholar] [CrossRef] - Jensen, A.; Pedersen, G.; Wood, D. An experimental study of wave run-up at a steep beach. J. Fluid Mech.
**2003**, 486, 161–188. [Google Scholar] [CrossRef] - Behrens, J. Numerical Methods in Support of Advanced Tsunami Early Warning. In Handbook of Geomathematics; Freeden, W., Nashed, M.Z., Sonar, T., Eds.; Springer Verlag: Heidelberg, Berlin, Germany, 2010; pp. 399–416. [Google Scholar]
- LeVeque, R.J.; George, D.L.; Berger, M.J. Tsunami modelling with adaptively refined finite volume methods. Acta Numer.
**2011**, 20, 211–289. [Google Scholar] [CrossRef] - Liu, P.L.F.; Yeh, H.H.J.; Synolakis, C. Advanced numerical models for simulating tsunami waves and run-up. In Advances in Coastal and Ocean Engineering; World Scientific Publishing Company Incorporated: Ithaca, NY, USA, 2008; Volume 10. [Google Scholar]
- Madsen, P.A.; Bingham, H.B.; Liu, H. A new Boussinesq method for fully nonlinear waves from shallow to deep water. J. Fluid Mech.
**2002**, 462, 1–30. [Google Scholar] [CrossRef] - Rakowsky, N.; Androsov, A.; Fuchs, A.; Harig, S.; Immerz, A.; Danilov, S.; Hiller, W.; Schröter, J. Operational tsunami modelling with TsunAWI—Recent developments and applications. Nat. Hazards Earth Syst. Sci.
**2013**, 13, 1629–1642. [Google Scholar] [CrossRef][Green Version] - Carrier, G.F.; Greenspan, H.P. Water waves of finite amplitude on a sloping beaching. J. Fluid Mech.
**1958**, 4, 97–109. [Google Scholar] [CrossRef] - Madsen, P.A.; Schäffer, H.A. Analytical solutions for tsunami runup on a plane beach: Single waves, N-waves and transient waves. J. Fluid Mech.
**2010**, 645, 27–57. [Google Scholar] [CrossRef] - Goring, D.G. Tsunamis—The Propagation of Long Waves onto a Shelf; Technical Report Caltech KHR: KH-R-38; California Institute of Technology: Pasadena, California, 1978. [Google Scholar]
- Synolakis, C.E.; Deb, M.K.; Skjelbreia, J.E. The anomalous behaviour of the runup of cnoidal waves. Phys. Fluids
**1988**, 31, 3–5. [Google Scholar] [CrossRef] - Liu, P.L.F.; Cho, Y.S.; Briggs, M.J.; Kanoglu, U.; Synolakis, C.E. Runup of solitary waves on a circular island. J. Fluid Mech.
**1995**, 302, 259–285. [Google Scholar] [CrossRef] - Madsen, P.A.; Fuhrman, D.R.; Schäffer, H.A. On the solitary wave paradigm for tsunamis. J. Geophys. Res.
**2008**, 113, C12012. [Google Scholar] [CrossRef] - Monaghan, J.J.; Kos, A. Scott Russel’s wave generator. Phys. Fluids
**2000**, 12, 622–630. [Google Scholar] [CrossRef] - Synolakis, C.E. The Runup of Long Waves. Ph.D. Thesis, California Institute of Technology, Pasadena, CA, USA, 1986. [Google Scholar]
- Synolakis, C.E. The runup of solitary waves. J. Fluid Mech.
**1987**, 185, 523–545. [Google Scholar] [CrossRef] - Titov, V.; Synolakis, C.E. Numerical Modeling of Tidal Wave Runup. J. Waterway Port Coastal Ocean Eng.
**1998**, 124, 157–171. [Google Scholar] [CrossRef] - Hammack, J.L. A note on tsunamis: Their generation and propagation in an ocean of uniform depth. J. Fluid Mech.
**1973**, 60, 769–800. [Google Scholar] [CrossRef] - Goseberg, N. A laboratory perspective of long wave generation. In Proceedings of the International Offshore and Polar Engineering Conference, Rhodes, Greece, 17–23 June 2012; pp. 54–60.
- Chanson, H.; Aoki, S.I.; Maruyama, M. An experimental study of tsunami runup on dry and wet horizontal coastlines. Sci. Tsunami Hazards
**2003**, 20, 278–293. [Google Scholar] - Goseberg, N.; Wurpts, A.; Schlurmann, T. Laboratory-scale generation of tsunami and long waves. Coastal Eng.
**2013**, 79, 57–74. [Google Scholar] [CrossRef] - Titov, V.; Gonzalez, F.J. Implementation and Testing of the Method of Splitting Tsunami (MOST) Model; NOAA Technical Memorandum ERL PMEL-112 1927; NOAA: Seattle, WA, USA, 1997.
- Hesthaven, J.S.; Warburton, T. Nodal Discontinuous Galerkin methods: Algorithms, Analysis, and Applications; Springer: New York, NY, USA, 2008. [Google Scholar]
- Giraldo, F.X.; Hesthaven, J.S.; Warburton, T. Nodal high-order discontinuous Galerkin methods for the spherical shallow water equations. J. Comput. Phys.
**2002**, 181, 499–525. [Google Scholar] [CrossRef][Green Version] - Kelly, J.; Giraldo, F. Continuous and Discontinuous Galerkin Methods for a Scalable 3D Nonhydrostatic Atmospheric Model: limited-area mode. J. Comput. Phys.
**2012**, 231, 7988–8008. [Google Scholar] [CrossRef][Green Version] - Synolakis, C.E.; Bernard, E.N.; Titov, V.V.; Kanoglu, U.; Gonzalez, F.I. Validation and Verification of Tsunami Numerical Models. Pure Appl. Geophys.
**2008**, 165, 2197–2228. [Google Scholar] [CrossRef] - Battjes, J.A. Surf similarity, 1974. In Proceedings of the 14th International Coastal Engineering Conference (ASCE), Copenhagen, Denmark, 24–28 June 1974.
- Goseberg, N. The Run-up of Long Waves—Laboratory-Scaled Geophysical Reproduction and Onshore Interaction with Macro-Roughness Elements. Ph.D. Thesis, The Leibniz University Hannover, Hannover, Germany, 2011. [Google Scholar]
- Goseberg, N. Reduction of maximum tsunami run-up due to the interaction with beachfront development—Application of single sinusoidal waves. Nat. Hazards Earth Syst. Sci.
**2013**, 13, 2991–3010. [Google Scholar] [CrossRef] - Goseberg, N.; Schlurmann, T. Non-stationary flow around buildings during run-up of tsunami waves on a plain beach. In Coastal Engineering Proceedings; Lynett, P., Ed.; World Scientific Publishing Company Incorporated: Seoul, South Korea, 2014; Volume 1. [Google Scholar]
- Goseberg, N.; Bremm, G.C.; Schlurmann, T.; Nistor, I. A transient approach flow acting on a square cylinder—Flow pattern and horizontal forces. In Proceedings of the 36th IAHR World Congress, Hague, The Netherlands, 28 June–3 July 2015; pp. 1–12.
- Bremm, G.C.; Goseberg, N.; Schlurmann, T.; Nistor, I. Long Wave Flow Interaction with a Single Square Structure on a Sloping Beach. J. Mar. Sci. Eng.
**2015**, 3, 821–844. [Google Scholar] [CrossRef] - Goseberg, N.; Schlurmann, T. Interaction of idealized urban infrastructure and long waves during run-up and on-land flow process in coastal regions. In Proceedings of the International Conference on Coastal Engineering; Lynett, P., Smith, J.M., Eds.; World Scientific Publishing Company Incorporated: Santander, Spain, 2012. [Google Scholar]
- Müller, D.R. Auflaufen und Überschwappen von Impulswellen an Talsperren. Ph.D. Thesis, Mitteilungen des Instituts, Versuchsanstalt für Wasserbau, Hydrologie und Glaziologie der Eidgenössischen Hochschule Zürich, Zürich, Germany, 1995. [Google Scholar]
- Heller, V. Scale effects in physical hydraulic engineering models. J. Hydraul. Res.
**2011**, 49, 293–306. [Google Scholar] [CrossRef] - Schüttrumpf, H. Wellenüberlaufströmung bei Seedeichen—Experimentelle und theoretische Untersuchungen. Ph.D. Thesis, Technische Universität Carolo-Wilhelmina, Braunschweig, Germany, 2001. [Google Scholar]
- Le Méhauté, B. An Introduction to Hydrodynamics & Water Waves; Springer: New York, NY, USA; Heidelberg, Berlin, Germany, 1976. [Google Scholar]
- Fuchs, H.; Hager, W.H. Scale effects of impulse wave run-up and run-over. J. Waterway Port Coastal Ocean Eng.
**2012**, 138, 303–311. [Google Scholar] [CrossRef] - Dingemann, M.W. Water wave propagation over uneven bottom. Part 1. Linear wave propagation. In Advanced Series on Ocean Engineering; World Scientific: Ithaca, NY, USA, 1997; Volume 13. [Google Scholar]
- Peakall, J.; Warburton, J. Surface tension in small hydraulic river models—The significance of the Weber number. J. Hydrology
**1996**, 53, 199–212. [Google Scholar] - Cockburn, B.; Lin, S.Y.; Shu, C.W. TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: One-dimensional systems. J. Comput. Phys.
**1989**, 84, 90–113. [Google Scholar] [CrossRef] - Vater, S.; Beisiegel, N.; Behrens, J. A Limiter-Based Well-Balanced Discontinuous Galerkin Method for Shallow-Water Flows with Wetting and Drying: One-Dimensional Case. Adv. Water Resour.
**2015**, 85, 1–13. [Google Scholar] [CrossRef] - Shu, C.W.; Osher, S. Efficient Implementation of Essentially Non-oscillatory Shock-Capturing Schemes. J. Comput. Phys.
**1988**, 77, 439–471. [Google Scholar] [CrossRef] - Gottlieb, S.; Shu, C.W.; Tadmor, E. Strong Stability-Preserving High-Order Time Discretization Methods. SIAM Rev.
**2001**, 43, 89–112. [Google Scholar] [CrossRef] - Courant, R.; Friedrichs, K.O.; Lewy, H. Über die partiellen Differenzengleichungen der mathematischen Physik. Math. Ann.
**1928**, 100, 32–74. [Google Scholar] [CrossRef] - Cockburn, B.; Shu, C.W. The Runge-Kutta Local Projection P
^{1}-Discontinuous-Galerkin Finite Element Method for Scalar Conservation Laws. RAIRO Modél. Math. Anal. Numér.**1991**, 25, 337–361. [Google Scholar] - Brier, G.W. Verification of forecasts expressed in terms of probability. Mon. Weather Rev.
**1950**, 78, 1–3. [Google Scholar] [CrossRef] - Madsen, P.A.; Fuhrman, D.R. Run-up of tsunamis and long waves in terms of surf-similarity. Coastal Eng.
**2008**, 55, 209–223. [Google Scholar] [CrossRef]

© 2015 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license ( http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**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