Applied Mathematics and Mechanics

, Volume 20, Issue 4, pp 343–349

# Hamiltonian formulation of nonlinear water waves in a two-fluid system

• Lu Dongqiang
• Dai Shiqiang
• Zhang Baoshan
Article

## Abstract

In this paper, it is dealt with that the Hamiltonian formulation of nonlinear water waves in a two-fluid system, which consists of two layers of constant-density incompressible inviscid fluid with a horizontal bottom, an interface and a free surface. The velocity potentials are expanded in power series of the vertical coordinate. By taking the kinetic thickness of lower fluid-layer and the reduced kinetic thickness of upper fluid-layer as the generalized displacements, choosing the velocity potentials at the interface and free surface as the generalized momenta and using Hamilton's principle, the Hamiltonian canonical equations for the system are derived with the Legendre transformation under the shallow water assumption. Hence the results for single-layer fluid are extended to the case of stratified fluid.

## Key words

two-fluid system Hamilton's principle nonlinear water waves shallow water assumption Hamiltonian canonical equations

## References

1. [1]
Luke J C. A variational principle for a fluid with a free surface [J].J Fluid Mech, 1967,27:395–397.
2. [2]
Whitham G B. Variational methods and application to water waves [J].Proc. Roy Soc A, 1967,299(1):6–25.
3. [3]
Zakharov V E. Stability of periodic wave of finite amplitude on the surface of a deep fluid [J].J Appl Mech Tech Phys, 1968,2(2):190–194.Google Scholar
4. [4]
Miles J W. On Hamilton's principle for surface waves [J].J Fluid Mech, 1977,83:153–158
5. [5]
Milder D M. A note regarding “On Hamilton's princilple for surfaces waves” [J].J Fluid Mech, 1977,83:159–161
6. [6]
Benjamin T B, Olver P J. Hamiltonian structure, symmetries and conservation laws for water wave [J].J Fluid Mech, 1982,125:137–185
7. [7]
Zhang Baoshan, Lu Dongqiang, Dai Shiqiang, et al.. Theory of Hamiltonian system for nonlinear waves and its applications [J].Advan in Mech, 1998,28 (4):521–531 (in Chinese)Google Scholar
8. [8]
Craig W, Groves M D. Hamiltonian long-wave approximations to the water waves problems [J].Wave Motion, 1994,19:367–389
9. [9]
Lu Dongqiang, Dai Shiqiang, Zhang Baoshan. Infinite-dimensional Harniltonian structure for nonlinear water waves [A]. In: Cheng Changjun, Dai Shiqiang, Liu Yulu eds.Modern Mathematics and Mechanics (MMM-VII) [C]. Shanghai: Shanghai University Press, 1997, 387–390 (in Chinese)Google Scholar
10. [10]
Dia Shiqiang. The interaction of two pairs of solitary waves is a two-fluid system [J].Scientia Sinica (Ser. A), 1984,27(5):507–520Google Scholar

© Editorial Committee of Applied Mathematics and Mechanics 1999

## Authors and Affiliations

• Lu Dongqiang
• 1
• Dai Shiqiang
• 1
• Zhang Baoshan
• 1
1. 1.Shanghai Institute of Applied Mathematics and MechanicsShanghai UniversityShanghaiP R China