Abhishek Sharma, Vijay G. Panchang


Accurate modeling of nonlinear wave transformation is important for studies related to harbor and nearshore design. While sophisticated finite-element models based on the elliptic mild-slope equation are often used for wave prediction, they do not include wave-wave interactions. These interactions, in general, involve transfer of energy and wave phase coupling among spectral components and are known to be quite significant especially in shoaling regions and inside harbors. To overcome this limitation, and to provide a basis for the eective modeling of nonlinear wave transformation in complex coastal and harbor environments, the development of a finite-element model based on the second-order nonlinear mild-slope equation of Kaihatu and Kirby (1995) is considered. The model uses an iterative procedure for solving the second-order boundary value problem. To ensure eective boundary treatment, a combination of frequently-used boundary conditions and the method of internal wave generation is used. Two cases involving nonlinear shoaling and harbor resonance are considered for model validation. Modeled results are compared with experimental data, and good agreement is observed in most cases. The methodology described in this paper can improve the applicability of existing finite-element based mild-slope models.


Nonlinear mild-slope equation; triad wave inteactions; harbor resonance

Full Text:



Berkhoff J.C.W., N. Booy, and A.C. Radder. 1982. Verification of numerical wave propagation models for simple harmonic linear water waves, Coastal Engineering, 6(3).

Berkhoff J.C.W. 1972. Computation of combined refraction-diffraction. Proceedings of 13th International Conference on Coastal Engineering, Vancouver (Canada) 471-490.

Kaihatu J.M., and J.T. Kirby. 1995. Nonlinear transformation of waves in finite water depth, Physics of Fluids, 7(8).

Kostense J.K., M.W. Dingemans, and P. van den Bosch. 1988 Wave-current interaction in harbours, In: Proceedings of 21th International Conference on Coastal Engineering., ASCE, New York, p. 32–46.

Li D., V.G. Panchang, Z. Tang, Z. Demirbilek, and J. Ramsden. 2005. Evaluation of an approximate method for incorporating floating docks in two-dimensional harbor wave prediction models, Canadian Journal of Civil Engineering, 32(3-4).

Liu P.L.F., S.B. Yoon, and J.T. Kirby. 1985. Nonlinear refraction-diffraction of waves in shallow water, Journal of Fluid Mechanics, 153(1).

Madsen P.A., and O.R. Sørensen. 1992. A new form of the Boussinesq equations with improved linear dispersion characteristics. part 2- a slowly-varying bathymetry, Coastal Engineering, 18(3-4).

Panchang V.G., and Z. Demirbilek. 2001. Simulation of waves in harbors using two-dimensional elliptic equation models, In: Advances in Coastal and Ocean Engineering Vol. 7. Word Scientific Publishing Co., p. 125–162.

Reddy J.N. 1993. An Introduction to The Finite Element Method, 2nd Edition, McGraw-Hill, USA.

Rogers S.R., and C.C. Mei. 1978. Nonlinear resonant excitation of a long and narrow bay, Journal of Fluid Mechanics, 88(1).

Sharma A., V.G. Panchang, and J.M. Kaihatu (2014). Modeling nonlinear wave-wave interactions with the elliptic mild slope equation, Applied Ocean Research, 48c, 114-125. (Accepted)

Tang Y., and Y. Ouellet (2014). A new kind of nonlinear mild-slope equation for combined refraction-diffraction of multifrequency waves, Coastal Engineering, 31, 3-36.

Whalin R.W. 1971. The limit of application of linear wave refraction theory in a convergence zone. Research Rept., Tech. Rep. H-71-3; USACE, Waterways Expt. Stn., Vicksburg, MS.

Woo S.B., and P.L.F. Liu. 2004. Finite-element model for modified Boussinesq equations. ii: Applications to nonlinear harbor oscillations, Journal of Waterways, Ports, Coastal and Ocean Engineering, 130(1).