Introduction of a Simple Cnoidal Wave Formulation Based on Nonlinear Interaction of Wave-Wave Principles


Iranian National Institute for Oceanography and Atmospheric Science (INIOAS)


In this study, a simple and efficient approach based on nonlinear wave interaction fundamentals is theoretically proposed to generate surface profile of the cnoidal waves. The approach includes Newton-Raphson algorithm to calculate the Ursell parameter and using a simple formulation. The wave profile resulted by means of introduced approach is determined as a superposing of limited number of cosine harmonics without encountering difficulties of using elliptic or hyperbolic functions, or any complex and complicated differential equations. It is demonstrated that a cnoidal wave profile is a result of high order self nonlinear interaction of primary frequency. Some definite energy is transmitted to higher harmonics due to nonlinear interactions. The amount of transmitted energy is controlled by Ursell parameter. The desirable accuracy determines the number of included harmonics in the proposed formulation and relative error of approach can be predicted based on Fourier and least square analysis techniques. The outputs of the proposed method are verified with cnoidal resulted from elliptic functions and the high efficiency of new approximation is revealed for engineering applications. The calculation of wave parameters such as energy flux, volume flux and radiation stress for cnoidal wave can be approximated using the proposed method. Using this approach, a physical interpretation of the Bm parameter (introduced in the first order of cnoidal wave theory) is presented. The calculation of several parameters such as velocity vectors and dynamic pressure duo to cnoidal waves is very simple by means of proposed approach.


1. Elgar S., Herbers T., Chandran V. and Guza R., (1995), Higher‐order spectral analysis of nonlinear ocean surface gravity waves. Journal of Geophysical Research: Oceans (1978–2012), Vol.100(C3):4977-83.
2. Phillips O., (1960), On the dynamics of unsteady gravity waves of finite amplitude Part 1. The elementary interactions. Journal of Fluid Mechanics, Vol.9(02):193-217.
3. Longuet-Higgins M. and Smith N., (1966), An experiment on third-order resonant wave interactions, Journal of Fluid Mechanics, Vol.25(03):417-35.
4. Beji S. and Battjes J., (1993), Experimental investigation of wave propagation over a bar, Coastal Engineering, Vol.19(1):151-62.
5. Dong G., Ma Y., Perlin M., Ma X., Yu B. and Xu J., (2008), Experimental study of wave–wave nonlinear interactions using the wavelet-based bicoherence, Coastal Engineering, Vol.55(9):741-52.
6. Elgar S., Freilich M. and Guza R., (1990), Model‐data comparisons of moments of nonbreaking shoaling surface gravity waves, Journal of Geophysical Research: Oceans (1978–2012), Vol.95(C9):16055-63.
7. Elgar S., Guza R. and Freilich M., (1993), Observations of nonlinear interactions in directionally spread shoaling surface gravity waves, Journal of Geophysical Research: Oceans, Vol.98(C11):20299-305.
8. Hasselmann K., Munk W. and MacDonald G., (1963), Bispectra of ocean waves. Time series analysis. p.p.:125-39.
9. Doering J.C. and Bowen A.J., (1987), Skewness in the nearshore zone: A comparison of estimates from Marsh‐McBirney current meters and colocated pressure sensors, Journal of Geophysical Research: Oceans (1978–2012). Vol.92(C12):13173-83.
10. Elgar S. and Guza R., (1985), Observations of bispectra of shoaling surface gravity waves, Journal of Fluid Mechanics, Vol.161:425-48.
11. Masuda A. and Kuo Y-Y., (1981), A note on the imaginary part of bispectra, Deep Sea Research Part A Oceanographic Research Papers, Vol.28(3):213-22.
12. Sénéchal N., Bonneton P. and Dupuis H., (2002), Field experiment on secondary wave generation on a barred beach and the consequent evolution of energy dissipation on the beach face, Coastal Engineering, Vol.46(3):233-47.
13. Mahmoudof S.M., Badiei P., Siadatmousavi S.M., Chegini V., (2016), Observing and estimating of intensive triad interaction occurrence in very shallow water, Continental Shelf Research, Vol.122:68-76.
14. Eldeberky Y., (2012), Nonlinear effects in gravity waves propagating in shallow water, Coastal Engineering Journal. Vol.54(04).
15. Armstrong J., Bloembergen N., Ducuing J. and Pershan P., (1962), Interactions between light waves in a nonlinear dielectric, Physical Review, Vol.127(6):1918.
16. Young I. and Eldeberky Y., (1998), Observations of triad coupling of finite depth wind waves, Coastal engineering, Vol.33(2):137-54.
17. Korteweg D.J. and De Vries G. XLI., (1895), On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, Vol.39(240):422-43.
18. Laitone E., (1960), The second approximation to cnoidal and solitary waves, Journal of Fluid Mechanics, Vol.9(03):430-44.
19. Monkmeyer P.L., (1970), A higher order theory for symmetrical gravity waves, 12th International Conference on Coastal Engineering, No. (12): 543-61.
20. Fenton J.D., (1986), Polynomial approximation and water waves, 20th International Conference on Coastal Engineering, No.(20):193-207.
21. Rienecker M. and Fenton J., (1981), Fourier approximation method for steady water waves, Journal of Fluid Mechanics, Vol.104(1):119.
22. Fenton J. and Gardiner-Garden R., (1982), Rapidly-convergent methods for evaluating elliptic integrals and theta and elliptic functions, The Journal of the Australian Mathematical Society Series B: Applied Mathematics, Vol.24(01):47-58.
23. Svendsen IA., (2006), Introduction to nearshore hydrodynamics, World Scientific. 24. Mansard E.P. and Funke E., (1980), The measurement of incident and reflected spectra using a least squares method, 17th International Conference on Coastal Engineering, No.(17): 154-72.