A third-order asymptotic solution of nonlinear standing water waves in Lagrangian coordinates
Chen Yang-Yih(陈阳益)a)† and Hsu Hung-Chu (许弘莒)b)‡
a Department of Marine Environment and Engineering, National Sun Yat-Sen University, Kaohsiung 804, Taiwan, China; b Tainan Hydraulics Laboratory, National Cheng Kung University, Tainan 701, Taiwan, China
Abstract Asymptotic solutions up to third-order which describe irrotational finite amplitude standing waves are derived in Lagrangian coordinates. The analytical Lagrangian solution that is uniformly valid for large times satisfies the irrotational condition and the pressure p=0 at the free surface, which is in contrast with the Eulerian solution existing under a residual pressure at the free surface due to Taylor's series expansion. In the third-order Lagrangian approximation, the explicit parametric equation and the Lagrangian wave frequency of water particles could be obtained. In particular, the Lagrangian mean level of a particle motion that is a function of vertical label is found as a part of the solution which is different from that in an Eulerian description. The dynamic properties of nonlinear standing waves in water of a finite depth, including particle trajectory, surface profile and wave pressure are investigated. It is also shown that the Lagrangian solution is superior to an Eulerian solution of the same order for describing the wave shape and the kinematics above the mean water level.
Received: 02 April 2008
Revised: 14 May 2008
Accepted manuscript online:
Chen Yang-Yih(陈阳益) and Hsu Hung-Chu (许弘莒) A third-order asymptotic solution of nonlinear standing water waves in Lagrangian coordinates 2009 Chin. Phys. B 18 861
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