# Obtaining Reflection Coefficients from a Single Point Velocity Measurement

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## Abstract

**:**

## 1. Introduction

## 2. Motivation

## 3. Theoretical Background

## 4. Numerical Experiment

## 5. Results

## 6. Discussion

## 7. Conclusions and Future Work

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- O’Boyle, L.; Elsäßer, B.; Whittaker, T. Methods to enhance the performance of a 3D coastal wave basin. Ocean Eng.
**2017**, 135, 158–169. [Google Scholar] [CrossRef] - Windt, C.; Davidson, J.; Schmitt, P.; Ringwood, J. Assessment of numerical wave makers. In Proceedings of the 12th European Wave and Tidal Energy Conference, Cork, Ireland, 27 August–1 September 2017. [Google Scholar]
- Thornton, E.B.; Calhoun, R.J. Spectral resolution of breakwater reflected waves. J. Waterw. Harbors Coast. Eng. Div.
**1972**, 98, 443–460. [Google Scholar] - Goda, Y.; Suzuki, Y. Estimation of incident and reflected waves in random wave experiments. In Proceedings of the 15th Coastal Engineering Conference, Honolulu, Hawaii, 1–17 July 1976; pp. 828–845. [Google Scholar]
- Mansard, E.P.D.; Funke, E.R. The measurement of incident and reflected spectra using a least squares method. In Proceedings of the 17th Coastal Engineering Conference, Sydney, Australia, 23–28 March 1980; pp. 154–172. [Google Scholar]
- Lin, C.Y.; Huang, C.J. Decomposition of incident and reflected higher harmonic waves using four wave gauges. Coast. Eng.
**2004**, 51, 395–406. [Google Scholar] [CrossRef] - Viviano, A.; Naty, S.; Foti, E.; Bruce, T.; Allsop, W.; Vicinanza, D. Large-scale experiments on the behaviour of a generalised Oscillating Water Column under random waves. Renew. Energy
**2016**, 99, 875–887. [Google Scholar] [CrossRef] [Green Version] - Isaacson, M. Measurement of regular wave reflection. J. Waterw. Port Coast. Ocean Eng.
**1991**, 117, 553–569. [Google Scholar] [CrossRef] - Frigaard, P.; Brorsen, M. A time-domain method for separating incident and reflected irregular waves. Coast. Eng.
**1995**, 24, 205–215. [Google Scholar] [CrossRef] [Green Version] - Ursell, F.; Dean, R.G.; Yu, Y.S. Forced small-amplitude water waves: A comparison of theory and experiment. J. Fluid Mech.
**1960**, 7, 33–52. [Google Scholar] [CrossRef] - Madsen, P.A. Wave reflection from a vertical permeable wave absorber. Coast. Eng.
**1983**, 7, 381–396. [Google Scholar] [CrossRef] - Baldock, T.; Simmonds, D. Separation of incident and reflected waves over sloping bathymetry. Coast. Eng.
**1999**, 38, 167–176. [Google Scholar] [CrossRef] - Brossard, J.; Hémonb, A.; Rivoalena, E. Improved analysis of regular gravity waves and coefficient of reflexion using one or two moving probes. Coast. Eng.
**2000**, 39, 193–212. [Google Scholar] [CrossRef] - Hughes, S.A. Laboratory wave reflection analysis using co-located gages. Coast. Eng.
**1993**, 20, 223–247. [Google Scholar] [CrossRef] - Hughes, S.A. Physical Models and Laboratory Techniques in Coastal Engineering; Advanced Series on Ocean Engineering; World Scientific: Singapore, 1993; Chapter 4; Volume 7. [Google Scholar]
- Neves, C.F.; Endres, L.A.M.; Fortes, C.J.; Clemente, D.S. The use of ADV in wave flumes: Getting more information about waves. Coast. Eng. Proc.
**2012**, 1, 38. [Google Scholar] [CrossRef] - Brede, H. Reflection Analysis of a new Gravel Beach Arrangement in Portaferry Wave Basin; Technical Report; Queen’s University Belfast: Belfast, UK, 2013. [Google Scholar]
- Lamont-Kane, P.; Folley, M.; Whittaker, T. Investigating uncertainties in physical testing of wave energy converter arrays. In Proceedings of the 10th European Wave and Tidal Energy Conference (EWTEC 2013), Aalborg, Denmark, 2–5 September 2013. [Google Scholar]
- Peric, R.; Abdel-Maksoud, M. Assessment of uncertainty due to wave reflections in experiments via numerical flow simulations. In Proceedings of the International Offshore and Polar Engineering Conference, Kona, HI, USA, 21–26 June 2015; pp. 530–537. [Google Scholar]
- Schmitt, P.; Elsaesser, B. A review of wave makers for 3D numerical simulations. In Proceedings of the Marine 2015 6th International Conference on Computational Methods in Marine Engineering, Rome, Italy, 15–17 June 2015; pp. 437–446. [Google Scholar]
- Schmitt, P.; Doherty, K.; Clabby, D.; Whittaker, T. The opportunities and limitations of using CFD in the development of wave energy converters. In Proceedings of the RINA Marine and Offshore Energy Conference, London, UK, 26–27 September 2012; pp. 89–97. [Google Scholar]
- Tezdogan, T.; Incecik, A.; Turan, O. Full-scale unsteady {RANS} simulations of vertical ship motions in shallow water. Ocean Eng.
**2016**, 123, 131–145. [Google Scholar] [CrossRef] - Vanneste, D.; Troch, P. 2D numerical simulation of large-scale physical model tests of wave interaction with a rubble-mound breakwater. Coast. Eng.
**2015**, 103, 22–41. [Google Scholar] [CrossRef] - Schmitt, P.; Elsäßer, B. On the use of OpenFOAM to model oscillating wave surge converters. Ocean Eng.
**2015**, 108, 98–104. [Google Scholar] [CrossRef] [Green Version] - Devolder, B.; Stratigaki, V.; Troch, P.; Rauwoens, P. CFD Simulations of Floating Point Absorber Wave Energy Converter Arrays Subjected to Regular Waves. Energies
**2018**, 11, 641. [Google Scholar] [CrossRef] - Olbert, G.; Schmitt, P. Steps towards fully nonlinear simulations of arrays of OWSC. In Proceedings of the 19th Numerical Towing Tank Symposium, St. Pierre d’Oleron, France, 3–4 October 2016. [Google Scholar]
- McKee, R.; Elsaesser, B.; O’Boyle, L. Investigating Tools used to Analyse the Impact of a Wave Farm on the surrounding Wave Field. In Proceedings of the 12th European Wave and Tidal Energy Conference, Cork, Ireland, 27 August–1 September 2017. [Google Scholar]
- O’Boyle, L.; Elsäßer, B.; Whittaker, T. Experimental Measurement of Wave Field Variations around Wave Energy Converter Arrays. Sustainability
**2017**, 9, 70. [Google Scholar] [CrossRef] - Dean, R.G.; Dalrymple, R.A. Water Wave Mechanics for Engineers and Scientists; Advanced Series on Ocean Engineering; World Scientific: Singapore, 1984; Chapter 4; Volume 2, pp. 78–130. [Google Scholar]
- Wallet, A.; Ruellan, F. Free-surface flow. In An Album of Fluid Motion; The Parabolic Press: Stanford, CA, USA, 1982; Chapter 7; pp. 110–111. [Google Scholar]
- Peric, R.; Abdel-Maksoud, M. Analytical prediction of reflection coefficients for wave absorbing layers in flow simulations of regular free-surface waves. Ocean Eng.
**2018**, 147, 132–147. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**Schematic representation of water particle velocities where h is the water depth and $\lambda $ is the wavelength for different water depths.

**Figure 2.**Schematic representation of the cross section of an incident and reflected wave in deep water where ${u}_{inc}$ and ${w}_{inc}$ are the horizontal and vertical velocity components for the incident wave, ${u}_{refl}$ and ${w}_{refl}$ are the horizontal and vertical velocity components for the reflected wave, ${u}_{res}$ and ${w}_{res}$ are the horizontal and vertical velocity components for the resultant wave, d is the distance of x to the reflective boundary, x is the horizontal distance, z is the vertical depth from the still water surface and $\theta $ is the tilt of the maxima axis to the horizontal axis.

**Figure 3.**Flow chart of the Orbital Velocity method where ${K}_{r}$ is the reflection coefficient, ${\theta}_{dataset}$ is the tilt of the ellipse of the dataset and $\left[k\widehat{x}\right]=(kx+\frac{\u03f5}{2})$.

Probe | X (m) | Y (m) | Z (m) |
---|---|---|---|

1 | 5.0 | 0.05 | 0.8 |

2 | 5.0 | 0.05 | 0.7 |

3 | 5.0 | 0.05 | 0.6 |

Probe | X (m) | Y (m) | Z (m) |
---|---|---|---|

1 | 5.0 | 0.05 | 0.2 |

2 | 5.0 | 0.05 | 0.15 |

3 | 5.0 | 0.05 | 0.1 |

Probe | X (m) |
---|---|

1 | 4.5 |

2 | 4.7 |

3 | 5.0 |

4 | 5.5 |

5 | 6.7 |

Wave Maker Number | Water Depth (m) | Wave Height (m) |
---|---|---|

1 | 0.0059 | |

2 | 1.0 | 0.0151 |

3 | 0.0274 | |

4 | 0.0090 | |

5 | 0.3 | 0.0179 |

6 | 0.0349 |

**Table 5.**Reflection coefficients using different methods from the deep water ( $1.0$ $\mathrm{m}$) simulations.

Reflection Coefficient | |||||
---|---|---|---|---|---|

Test Case | Wave Makers | Predicted | Orbital Velocity | Mansard and Funke Method | |

Method | Combination 1 | Combination 2 | |||

A | 1 and 3 | 0.22 | 0.27 | 0.25 | 0.26 |

B | 2 and 3 | 0.55 | 0.52 | 0.52 | 0.51 |

C | 3 and 3 | 1.00 | 0.97 | 0.95 | 0.99 |

**Table 6.**Reflection coefficients using different methods from the intermediate water ( $0.3$ $\mathrm{m}$) simulations.

Reflection Coefficient | |||||
---|---|---|---|---|---|

Test Case | Wave Makers | Predicted | Orbital Velocity | Mansard and Funke Method | |

Method | Combination 1 | Combination 2 | |||

D | 4 and 6 | 0.26 | 0.26 | 0.25 | 0.23 |

E | 5 and 6 | 0.51 | 0.53 | 0.53 | 0.54 |

F | 6 and 6 | 1.00 | 0.98 | 1.06 | 0.97 |

**Table 7.**Reflection coefficients for probes of varying vertical positions in deep water ( 1 $\mathrm{m}$) simulations.

Test Case | ||||||
---|---|---|---|---|---|---|

Distance from Seabed (m) | A | B | C | |||

${\mathit{K}}_{\mathit{r}}$ | Error | ${\mathit{K}}_{\mathit{r}}$ | Error | ${\mathit{K}}_{\mathit{r}}$ | Error | |

0.8 | 0.27 | 0.52 | 0.97 | |||

0.7 | 0.28 | 3.1% | 0.54 | 3.8% | 0.96 | −1% |

0.6 | 0.27 | 0% | 0.52 | 0% | 0.98 | 1% |

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**MDPI and ACS Style**

McKee, R.; Elsäßer, B.; Schmitt, P.
Obtaining Reflection Coefficients from a Single Point Velocity Measurement. *J. Mar. Sci. Eng.* **2018**, *6*, 72.
https://doi.org/10.3390/jmse6020072

**AMA Style**

McKee R, Elsäßer B, Schmitt P.
Obtaining Reflection Coefficients from a Single Point Velocity Measurement. *Journal of Marine Science and Engineering*. 2018; 6(2):72.
https://doi.org/10.3390/jmse6020072

**Chicago/Turabian Style**

McKee, Rachael, Björn Elsäßer, and Pál Schmitt.
2018. "Obtaining Reflection Coefficients from a Single Point Velocity Measurement" *Journal of Marine Science and Engineering* 6, no. 2: 72.
https://doi.org/10.3390/jmse6020072