An Analytical Solution for Regular Progressive Water Waves

Title & Authors
An Analytical Solution for Regular Progressive Water Waves
Shin, JangRyong;

Abstract
In order to provide simple and accurate wave theory in design of offshore structure, an analytical approximation is introduced in this paper. The solution is limited to flat bottom having a constant water depth. Water is considered as inviscid, incompressible and irrotational. The solution satisfies the continuity equation, bottom boundary condition and non-linear kinematic free surface boundary condition exactly. Error for dynamic condition is quite small. The solution is suitable in description of breaking waves. The solution is presented with closed form and dispersion relation is also presented with closed form. In the last century, there have been two main approaches to the nonlinear problems. One of these is perturbation method. Stokes wave and Cnoidal wave are based on the method. The other is numerical method. Dean`s stream function theory is based on the method. In this paper, power series method was considered. The power series method can be applied to certain nonlinear differential equations (initial value problems). The series coefficients are specified by a nonlinear recurrence inherited from the differential equation. Because the non-linear wave problem is a boundary value problem, the power series method cannot be applied to the problem in general. But finite number of coefficients is necessary to describe the wave profile, truncated power series is enough. Therefore the power series method can be applied to the problem. In this case, the series coefficients are specified by a set of equations instead of recurrence. By using the set of equations, the nonlinear wave problem has been solved in this paper.
Keywords
Nonlinear progressive water waves;Breaking waves;Breaking limit;Non-linear free surface boundary condition;The Stokes criterion;The power series method;The Fourier series method;Variational method;
Language
English
Cited by
1.
Analytical Approximation in Deep Water Waves,;

Journal of Advanced Research in Ocean Engineering, 2016. vol.2. 1, pp.1-11
1.
Analytical Approximation in Deep Water Waves, Journal of Advanced Research in Ocean Engineering, 2016, 2, 1, 1
References
1.
Chakrabarti, S, K., "Hydrodynamics of Offshore Structures," WIT press, First Edition (1987).

2.
Dean, R, G., "Stream function representation of nonlinear ocean waves," Journal of Geophysical Research, Vol 70, No 18, pp 4561-4572 (1965).

3.
Det Norske Veritas, "Environmental conditions and environmental loads," DNV-RP-C205, (2007).

4.
Dym, Clive, L and Shames Irving, H, " Solid mechanics : A Variational Approach," McGraw-Hill Series in Advanced Engineering, First Edition, pp 109-173 (1973).

5.
Fenton, J, D, and Rienecker, M, M, "A Fourier method for solving nonlinear water-wave problems: application to solitary-wave interations," J. Fluid Mech, Vol 118, pp 411-443 (1982).