Volume no :11, Issue no: 1, February

CNOIDAL WAVE SOLUTIONS IN SHALLOW WATER AND SOLITARY WAVE LIMIT

Author's: Shahana Parvin, M. Shamsul Alam Sarker and Shamima Sultana
Pages: [63] - [93]
Received Date: December 3, 2013; Revised January 5, 2014
Submitted by:

Abstract

Equations of motion in frame moving with the waves are considered for steady, incompressible flow. The boundary conditions at the bottom and at the free surface are used for solving shallow water wave problems. The remaining boundary conditions are also taken from Navier-Stokes equation of motion. Using these boundary conditions, three nonlinear ordinary differential equations are formulated, which can be solved by using series expansion method. We consider that all variations in X is relatively slow and can be expressed in terms of dimensionless variable where is a small quantity and h is the trough depth of fluid. Then approaching on series expansion method, two types of nonlinear ordinary differential equations are formulated. Using Jacobi elliptic function, first and second order cnoidal wave solutions have been derived. Then mean value of Jacobi elliptic function and the solitary wave limit of cnoidal wave solutions are also formulated.

Keywords

Navier-Stokes equation, Jacobi elliptic function, cnoidal wave, solitary wave limit.