Abstract
Long waves such as tsunamis can be trapped by islands due to wave refraction, and these trapped waves will cause huge damage even in the sheltered shoreline of the island. That all waves propagating into the topography and finally reaching the coastline are called perfect trapped modes, while any waves escaping from the topography are called leaky modes. Whether these long waves can be trapped is dependent on the depth profile of the island. This paper presents analytic solutions of the ray path for waves propagating into the circular island with power function profiles. Wave height distributions over the island are further investigated based on the principia that crowded rays correspond to large wave height and sparse rays correspond to small wave height. The trapped mechanism for water waves over the island is revealed based on their ray paths. Furthermore, the perfectly trapped criterion is derived, that is, when the slope gradient at the topography toe is greater than twice the ratio of the water depth to the radial distances, all wave rays propagating on the island will finally reach the coastline, and the waves are perfectly trapped.
Similar content being viewed by others
References
Anca-Voichita, M., 2012. An exact solution for geophysical equatorial edge waves over a sloping beach. Journal of Physics A: Mathematical and Theoretical, 45 (36): 365501.
Fine, I. V., Kulikov, E. A., and Cherniawsky, J. Y., 2013. Japan’s 2011 tsunami: Characteristics of wave propagation from observations and numerical modelling. Pure and Applied Geophysics, 170 (6): 1295–1307.
Ionescu-Kruse, D., 2014. Instability of edge waves along a sloping beach. Journal of Differential Equations, 256 (12): 3999–4012.
Jung, T. H., Lee, C., and Cho, Y. S., 2010. Analytical solutions for long waves over a circular island. Coastal Engineering, 57 (4): 440–446.
Koshimura, S.-I., Imamura, F., and Shuto, N., 2001. Characteristics of tsunamis propagating over oceanic ridges: Numerical simulation of the 1996 Irian Jaya earthquake tsunami. Natural Hazards, 24 (3): 213–229.
Kowalik, Z., Knight, W., Logan, T., and Whitmore, P., 2005. Numerical modeling of the global tsunami: Indonesian Tsunami of 26 December 2004. Science of Tsunami Hazards, 23 (1): 40–56.
Longuet-Higgins, M. S., 1967. On the trapping of wave energy round islands. Journal of Fluid Mechanics, 29 (4): 781–821.
Longuet-Higgins, M. S., 1969. On the trapping of long-period waves round islands. Journal of Fluid Mechanics, 37 (4): 773–784.
McFall, B. C., and Fritz, H. M., 2016. Physical modelling of tsunamis generated by three-dimensional deformable granular landslides on planar and conical island slopes. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science, 472: 20160052.
Mei, C. C., and Angelides, D., 1977. Longshore circulation around a conical island. Coastal Engineering, 1 (1): 31–42.
Sato, S., 1996. Numerical simulation of 1993 southwest Hokkaido earthquake tsunami around Okushiri Island. Journal of Waterway, Port, Coastal, and Ocean Engineering, 122 (5): 209–215.
Shaw, R. P., and Neu, W. L., 1988. Long wave trapping by axisymmetric topographies. In: Natural and Man-Made Hazards. El-Sabh, M. I., and Murty, T. S., eds., Springer Netherlands, Dordrecht, 227–238.
Shuto, N., 2007. Damage and reconstruction at Okushiri Town caused by the 1993 Hokkaido Nansei-Oki Earthquake Tsunami. Journal of Disaster Research, 2 (1): 44–49.
Snodgrass, F. E., Munk, W. H., and Miller, G. R., 1962. Long period waves over California’s continental borderland. Journal of Marine Research, 20: 3–30.
Stokes, G. G., 1846. Report on recent researches in hydrodynamics. British Association for the Advancement of Science Report. 1: 1–20.
Summerfield, W., 1972. Circular islands as resonators of longwave energy. Philosophical Transactions of the Royal Society A: Mathematical Physical and Engineering Sciences, 272 (1225): 361–402.
Tinti, S., and Vannini, C., 1995. Tsunami trapping near circular islands. Pure and Applied Geophysics, 144: 595–619.
Ursell, F., 1952. Edge waves on a sloping beach. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences,, 214 (1116): 79–97.
Yeh, H., Liu, P., Briggs, M. and Synolakis, C., 1994. Propagation and amplification of tsunamis at coastal boundaries. Nature, 372 (6504): 353–355.
Yu, X. and Zhang, B., 2003. An extended analytic solution for combined refraction and diffraction of long waves over circular shoals. Ocean Engineering, 30 (10): 1253–1267.
Zhai, X. Y., Liu, H. W., and Xie, J. J., 2013. Analytic study to wave scattering by a general Homma island using the explicit modified mild-slope equation. Applied Ocean Research, 43 (10): 175–183.
Zhang, Y., and Zhu, S., 1994. New solutions for the propagation of long water waves over variable depth. Journal of Fluid Mechanics, 278: 391–406.
Zheng, J. H., Xiong, M. J., and Wang, G., 2016. Trapping mechanism of submerged ridge on trans-oceanic tsunami propagation. China Ocean Engineering, 30: 271–282.
Acknowledgements
This research was supported by the National Key Research and Development Program of China (No. 2016YFC 1402800), the National Science Fund for Distinguished Young Scholars (No. 51425901), the National Natural Science Foundation of China (No. 51579090), and Innovation Project of Colleges and Universities in Jiangsu Province (No. 2015B41814).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Zheng, J., Fu, D. & Wang, G. Trapping mechanism for long waves over circular islands with power function profiles. J. Ocean Univ. China 16, 655–660 (2017). https://doi.org/10.1007/s11802-017-3404-7
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11802-017-3404-7