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Mathematical Problems in Engineering
Volume 2013 (2013), Article ID 253401, 12 pages
Research Article

On the Study of Second-Order Wave Theory and Its Convergence for a Two-Fluid System

1Division of Mathematics, General Education Center, Chienkuo Technology University, Changhua city 500, Taiwan
2International Wave Dynamics Research Center, National Cheng Kung University, Tainan 701, Taiwan
3Department of Hydraulic and Ocean Engineering, National Cheng Kung University, Tainan 701, Taiwan
4Tainan Hydraulic Laboratory, National Cheng Kung University, Tainan 701, Taiwan

Received 24 December 2012; Accepted 19 March 2013

Academic Editor: Kuppalapalle Vajravelu

Copyright © 2013 Chi-Min Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Second-order solutions of internal and surface waves in a two-fluid system are theoretically analyzed in this study. Using the perturbation technique, the derivation of second-order solutions for internal waves is revisited, and the results are expressed in one-by-one forms instead of a matrix form. Second-order solutions arising from the interactions of two arbitrary linear waves of different frequencies contain the sum-frequency (superharmonic) and the difference-frequency (subharmonic) components, which are separately examined. Internal Stokes wave being a special case of present solutions is firstly investigated. Next, the convergence of second-order theory and the second-order effects on wave profiles are analyzed. For general cases, the effects of the thickness ratio of two fluids and the ratio of wavenumbers of two first-order waves on second-order wave characteristics, which include transfer functions and particle velocities, are also examined. Moreover, most existing theories for the one-fluid and two-fluid systems can be deduced from present solutions.