Multiscale Mechanisms of Water-Wave Energy Focusing by Reflective Geometries

Abstract

Reflectors represent an effective strategy for enhancing the regional energy density of water waves. The distinct characteristics of water waves from optics need to be clarified to promote the application of this technology. In this work, three reflector geometries—comprising one linear type (straight) and two nonlinear profiles (semi-circular, parabolic)—are examined to evaluate their respective wave-focusing effects. Three numerical approaches—one-dimensional (1D) ray-tracing theory, two-dimensional (2D) wave equations, and three-dimensional (3D) Navier-Stokes equations—were employed to compute the reflection characteristics over a broad range of wavelengths. The results indicate that the reflection behavior of a nonlinear reflector can generally be separated into three distinct regimes: short waves, intermediate waves, and long waves. In the short-wave regime, wave focusing can be approximated as the superposition of ray trajectories. For intermediate waves, complex interference among wave components within a wavelength must be considered. In the long-wave regime, the nonlinear reflector effectively acts as a “straight reflector” in the far-field view. Moreover, the introduced 1D and 2D models offer an efficient way to predict the energy-focusing laws of the reflector, whereas the 3D model provides more accurate wave-field details.

References

[1] Aumann HM, Russell T, Emanetoglu NW. Comparison of a small parabolic reflector for use with an acoustic and a radar microphone. Proceedings of the IEEE International Symposium on Antennas and Propagation and USNC/URSI National Radio Science Meeting, 2015, Piscataway, USA: IEEE, 2199-2200

[2] Bahadori K, Rahmat-Samii Y. Characterization of effects of periodic and aperiodic surface distortions on membrane reflector antennas. IEEE Transactions on Antennas and Propagation, 2005, 53927822791.

[3] Bobinski T, Eddi A, Petitjeans P, Maurel A, Pagneux V. Experimental demonstration of epsilon-near-zero waterwaves focusing. Applied Physics Letters, 2015, 107: 014101.

[4] Cheng Y, Xi C, Dai S, Ji C, Cocard M, Yuan Z, Incecik A. Performance characteristics and parametric analysis of a novel multi-purpose platform combining a moonpool-type floating breakwater and an array of wave energy converters. Applied Energy, 2021, 292: 116888.

[5] Chung FF, Ong MC, Wang J. Focusing monochromatic water surface waves by manipulating the phases using submerged blocks. Journal of Marine Science and Engineering, 2024, 12101706.

[6] Cong P, Ge M, Zhou F, Ning D. Layout optimization of oscillating water column (OWC) wave energy converter (WEC) array coupled with a parabolic-wall energy concentrator. Energy, 2025, 340: 139207.

[7] Griffiths LS, Porter R. Focusing of surface waves by variable bathymetry. Applied Ocean Research, 2012, 34: 150-163.

[8] Heo S, Koo W, Lee OB. Optimization analysis of a heaving point absorber wave energy converter in wave focusing conditions near a parabolic vertical wall. Ocean Engineering, 2025, 331: 121342.

[9] Howe D, Nader J. OWC WEC integrated within a breakwater versus isolated: Experimental and numerical theoretical study. International Journal of Marine Energy, 2017, 20: 165-182.

[10] Hu X, Chan CT. Refraction of water waves by periodic cylinder arrays. Physical Review Letters, 2005, 9515154501.

[11] Jun SE, Yoon SH, Kim YH. Enhancing acoustical quality in a circular-shaped music hall by combining field measurements and computer simulation methodology. Journal of Building Engineering, 2025, 113: 114091.

[12] Kara F. Hydrodynamic performances of wave energy converter arrays in front of a vertical wall. Ocean Engineering, 2021, 235: 109459.

[13] Kara F. Effects of a vertical wall on wave power absorption with wave energy converters arrays. Renewable Energy, 2022, 196: 812-823.

[14] Kong F, Liu H, Su W, Ao J, Chen H, Jing F. Analytical and numerical analysis of the dynamics of a moonpool platform-wave energy buoy (MP-WEB). Energies, 2019, 12214083.

[15] Konispoliatis DN. Performance of an array of oscillating water column devices in front of a fixed vertical breakwater. Journal of Marine Science and Engineering, 2020, 811912.

[16] Konispoliatis DN. Assessment of the hydrodynamic performance of an oscillating water column device in front of a V-shaped vertical wall. Journal of Offshore Mechanics and Arctic Engineering, 2023, 1455052001.

[17] Konispoliatis DN, Chatjigeorgiou IK, Mavrakos SA. Near trapped wave phenomena in an array of truncated cylinders in a perpendicular arrangement in front of a vertical breakwater. Applied Mathematical Modelling, 2020, 83: 497-525.

[18] Konispoliatis DN, Mavrakos SA. Theoretical performance investigation of a vertical cylindrical oscillating water column device in front of a vertical breakwater. Journal of Ocean Engineering and Marine Energy, 2020, 6: 1-13.

[19] Konispoliatis DN, Mavrakos SA. Wave power absorption by arrays of wave energy converters in front of a vertical breakwater: A theoretical study. Energies, 2020, 1381985.

[20] Konispoliatis DN, Mavrakos SA. Hydrodynamic efficiency of a wave energy converter in front of an orthogonal breakwater. Journal of Marine Science and Engineering, 2021, 9194.

[21] Konispoliatis DN, Mavrakos SA, Katsaounis GM. Theoretical evaluation of the hydrodynamic characteristics of arrays of vertical axisymmetric floaters of arbitrary shape in front of a vertical breakwater. Journal of Marine Science and Engineering, 2020, 8162.

[22] Kudo K, Tsuzuku T, Imai K, Akiyama Y. Study on wave focusing by a horizontally submerged plate. Journal of the Society of Naval Architects of Japan, 1986, 160: 217-225. in Japanese)

[23] Liu G, Mayon R, Sergiienko N, Ning D. Point absorber wave energy converter arrays coupled with a V-shaped breakwater energy concentrator. Ocean Engineering, 2025, 342: 123050.

[24] Liu H, Yan F, Jing F, Ao J, Han Z, Kong F. Numerical and experimental investigation on a moonpool-buoy wave energy converter. Energies, 2020, 1392364.

[25] Mavrakos SA, Katsaounis GM, Nielsen K, Lemonis G. Numerical performance investigation of an array of heaving wave power converters in front of a vertical breakwater. Proceedings of the 14th International Offshore and Polar Engineering Conference, Toulon, France, 2004238245

[26] Mayon R, Ning D, Sun Y, Ding Z, Wang R, Zhou Y. Experimental investigation on a novel and hyper-efficient oscillating water column wave energy converter coupled with a parabolic breakwater. Coastal Engineering, 2023, 185: 104360.

[27] Mayon R, Ning D, Xu J, Fu L. Oscillating water column wave energy converter arrays coupled with a parabolic-wall energy concentrator in regular and irregular wave conditions. Coastal Engineering, 2024, 192: 104559.

[28] Mayon R, Ning DZ, Zhang C, Chen L, Wang R. Wave energy capture by an omnidirectional point sink oscillating water column system. Applied Energy, 2021, 304: 117795.

[29] McIver P, Evans DV. An approximate theory for the performance of a number of wave-energy devices set into a reflecting wall. Applied Ocean Research, 1988, 10: 58-65.

[30] Mehlum E, Stamnes J. On the focusing of ocean swells and its significance in power production, 1978, Oslo, Norway: Central Institute for Industrial Research, 1-3877

[31] Mehlum E, Stamnes J. Power production based on focusing of ocean swells. Proceedings of the First Symposium on Wave Energy Utilization, Gothenburg, Sweden, 19792935

[32] Mei CC, Stiassnie M, Yue DKP. Theory and applications of ocean surface waves, 2005, Singapore: World Scientific

[33] Miao J, Guo Z, Zhai Z. Scattering of water waves by multiple segmented concentric semi-closed breakwaters and an interior cylinder: An analytical approach. Physics of Fluids, 2023, 35: 126609.

[34] Murashige S, Kinoshita T. Fundamental studies on ocean wave focusing. Journal of the Society of Naval Architects of Japan, 1990, 168: 183-192. in Japanese)

[35] Murashige S, Kinoshita T. An ideal ocean wave focusing lens and its shape. Applied Ocean Research, 1992, 14: 275-290.

[36] Natarajan SK, Cho IH. Cost-effective optimization of an array of wave energy converters in front of a verticalseawall. Energies, 2024, 171128.

[37] Ning DZ, Zhao XL, Chen LF, Zhao M. Hydrodynamic performance of an array of wave energy converters integrated with a pontoon-type breakwater. Energies, 2018, 113685.

[38] Ottley M. Designing for speech in a circular room. Proceedings of ACOUSTICS 2018: Hear to Listen, Adelaide, Australia, 2018100112

[39] Rahmat-Samii Y, Densmore A. A history of reflector antennas development past, present and future. Proceedings of the SBMO/IEEE MTT-S International Microwave and Optoelectronics Conference, Belém, Brazil, 20091723

[40] Ren J, Jin P, Liu Y, Zang J. Wave attenuation and focusing by a parabolic arc pontoon breakwater. Energy, 2021, 217: 119405.

[41] Sawant O, Bhowmick A, Sreenivas TV. Enhanced directional sensitivity using acoustic dish reflector. Proceedings of the International Conference on Signal Processing and Communications (SPCOM), Bangalore, India, 2018162166

[42] Schay J, Bhattacharjee J, Guedes Soares C. Numerical modelling of a heaving point absorber in front of a vertical wall. Proceedings of the 32nd International Conference on Ocean, Offshore and Arctic Engineering, Nantes, France, 2013V008T09A097

[43] Stamnes JJ, Lovhaugen O, Spjelkavik B, Mei CC, Lo E, Yue DKP. Nonlinear focusing of surface waves by a lens-theory and experiment. Journal of Fluid Mechanics, 1983, 135: 71-94.

[44] van der Wiel RJ, Kramer J, van der Ven PPD, Borsboom MJA, de Jong MPC. Guedes Soares C. Influence of a parabolic reflector wall on the sea state in an array of point absorber wave energy converters. Progress in Renewable Energies Offshore, 2016, Boca Raton, USA: CRC Press, 393-404.

[45] Wang Z, Zhang P, Nie X, Zhang Y. Manipulating water wave propagation via gradient index media. Scientific Reports, 2015, 5: 16846.

[46] Wolf KB. Geometric optics on phase space, 2004, Berlin, Germany: Springer

[47] Xu J, Ning D, Mayon R, Zhao M. Hydrodynamic investigation of a parabolic breakwater for wave energy focusing. Physics of Fluids, 2023, 35: 097145.

[48] Yilmaz i, Mwesigye A. Modeling, simulation and performance analysis of parabolic trough solar collectors: A comprehensive review. Applied Energy, 2018, 225: 135-174.

[49] Zhai Z, Hu Q, Ye W, Huang H. Analytical modelling of solitary wave diffraction from a V-shaped breakwater. Ocean Engineering, 2021, 230: 109014.

[50] Zhai Z, Shao Y, Wang K, Huang H, Li H. Semi-analytical solution of cnoidal wave diffraction around a double-layer arc-shaped vertical porous breakwater. Journal of Fluids and Structures, 2021, 103: 103261.

[51] Zhang C, Ning D. Hydrodynamic study of a novel breakwater with parabolic openings for wave energy harvest. Ocean Engineering, 2019, 182: 540-551.

[52] Zhang SX, Duan BY, Bao H, Lian PY. Sensitivity analysis of reflector antennas and its application on shaped geo-truss unfurlable antennas. IEEE Transactions on Antennas and Propagation, 2013, 611154025407.

[53] Zhang Z, Liu S, Luan Z, Wang Z, He G. Invisibility concentrator for water waves. Physics of Fluids, 2020, 328081701.

[54] Zhao XL, Ning DZ, Liang DF. Experimental investigation on hydrodynamic performance of a breakwater-integrated WEC system. Ocean Engineering, 2019, 171: 25-32.

[55] Zhao XL, Ning DZ, Zhang CW, Liu YY, Kang HG. Analytical study on an oscillating buoy wave energy converter integrated into a fixed box-type breakwater. Mathematical Problems in Engineering, 2017, 201713960401.

[56] Zhou B, Wang Y, Zheng Z, Jin P, Ning D. Power generation and wave attenuation of a hybrid system involving a heaving cylindrical wave energy converter in front of a parabolic breakwater. Energy, 2023, 282: 128364.

[57] Zhou B, Zheng Z, Zhang Q, Jin P, Wang L, Ning D. Wave attenuation and amplification by an abreast pair of floating parabolic breakwaters. Energy, 2023, 271: 127077.

[58] Zong Y, Duan B, Ban Y, Wang W, Xu W, Wang C, Du J. Surface configuration design of cable-network reflectors considering the radiation pattern. IEEE Transactions on Antennas and Propagation, 2014, 62631633173.

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