Abstract
A novel theoretical approach is applied to predict the propagation and transformation of transient nonlinear waves on a current. The problem was solved by applying an eigenfunction expansion method and the derived semi-analytical solution was employed to study the transformation of wave profile and the evolution of wave spectrum arising from the nonlinear interactions of wave components in a wave train which may lead to the formation of very large waves. The results show that the propagation of wave trains is significantly affected by a current. A relatively small current may substantially affect wave train components and the wave train shape. This is observed for both opposing and following current. The results demonstrate that the application of the nonlinear model has a substantial effect on the shape of a wave spectrum. A train of originally linear and very narrow-banded waves changes its one-peak spectrum to a multi-peak one in a fairly short distance from an initial position. The discrepancies between the wave trains predicted by applying the linear and nonlinear models increase with the increasing wavelength and become significant in shallow water even for waves with low steepness. Laboratory experiments were conducted in a wave flume to verify theoretical results. The free-surface elevations recorded by a system of wave gauges are compared with the results provided by the nonlinear model. Additional verification was achieved by applying a Fourier analysis and comparing wave amplitude spectra obtained from theoretical results with experimental data. A reasonable agreement between theoretical results and experimental data is observed for both amplitudes and phases. The model predicts fairly well multi-peak spectra, including wave spectra with significant nonlinear wave components.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Baddour, R. E. and Song, S., 1990. On the interaction between waves and currents, Ocean Eng., 17(1–2): 1–21.
Bretherton, F. P. and Garrett, C. J. R., 1968. Wavetrains in inhomogeneous moving media, Proc. R. Soc. Lond., A, 302, 529–554.
Chen, Q., Madsen, P. A., Schäffer, H. A. and Basco, D. R., 1998. Wave-current interaction based on an enhanced Boussinesq approach, Coast. Eng., 33(1): 11–39.
Chawla, A. and Kirby, J. T., 2002. Monochromatic and random wave breaking at blocking points, J. Geophys. Res., 107(C7): 41–49.
Dean, R. G. and Dalrymple, R. T., 1984. Water Wave Mechanics for Engineers and Scientists, Prentice-Hall.
Fenton, J. D., 1999. Numerical methods for nonlinear waves, in: Advances in Coastal and Ocean Engineering, Liu, P. L-F, editor, 5, World Scientific.
Kinsman, B., 1965. Wind Waves, Prentice-Hall Inc., Englewood Cliffs, New Jersey.
Ma, Y. X., Ma, Y. Z., Dong, G. H., Liu, S. X. and Li, J. X., 2009. Numerical study of influence of currents on nonlinear focusing waves, China Ocean Eng., 23(4): 623–634.
Madsen, P. A. and Schäffer, H. A., 1998. Higher-order Boussinesq-type equations for surface gravity, Phil. Trans. R. Soc. Lond., A, 356, 3123–3181.
Peregrine, D. H. and Smith, R., 1979. Nonlinear effects upon waves near caustics, Phil. Trans. R. Soc. Lond., A, 292, 341–370.
Press, W. H., Flannery, B., Teukolsky, S. A. and Vetterling, W. T., 1988. Numerical Recipes, Cambridge University Press, Cambridge.
Suastika, I. A. and Battjes, J. K., 2009. A model for blocking of periodic waves, Coast. Eng. J., 51(2): 81–99.
Sulisz, W. and Hudspeth, R. T., 1993. Complete second-order solution for water waves generated in wave flumes, J. Fluid. Struct., 7(3): 253–268.
Sulisz, W., 2003. Numerical modeling of wave absorbers for physical wave tanks, J. Waterw. Port Coast. Ocean Eng., 129(1): 5–14.
Sulisz, W. and Paprota, M., 2004. Modeling of the propagation of transient waves of moderate steepness, Appl. Ocean Res., 26(3–4): 137–146.
Sulisz, W. and Paprota, M., 2008. Generation and propagation of transient nonlinear waves in a wave flume, Coast. Eng., 55(4): 277–287.
Teng, B., Zhao. M. and Bai, W., 2001. Wave diffraction in a current over a local shoal, Coast. Eng., 42(2): 163–172.
Wehausen, J. V., 1960. Surface Waves, In Handbuch der Physik, 9, Springer-Verlag, Berlin, 446–778.
Yoon, S. B. and Liu, P. L. -F., 1989. Interaction of current and weakly nonlinear water waves in shallow water, J. Fluid Mech., 205, 397–419.
Author information
Authors and Affiliations
Corresponding author
Additional information
This paper was supported partially by the Institute of Hydroengineering of the Polish Academy of Sciences and the state budget for research for the years 2010–2011.
Rights and permissions
About this article
Cite this article
Sulisz, W., Paprota, M. Modeling of propagation and transformation of transient nonlinear waves on a current. China Ocean Eng 27, 579–592 (2013). https://doi.org/10.1007/s13344-013-0049-1
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13344-013-0049-1