Contribution of the Second Order Terms to the Nonlinear Shallow Water Waves

Contribution of the second order terms in the reductive perturbation theory has been investigated for the nonlinear shallow water waves. The fundamental equations are reduced to a coupled set of the Korteweg-de Vries equation for the first order horizontal velocity and a linear inhomogeneous equation for the second order arbitrary function. Structure of the coupled set of equations turns out to be the same as in the case of nonlinear ion acoustic wave. A steady state solution of the coupled set of equations has been examined in comparison with Laiton's analysis of the second order contribution of the Friedrich's expansion for the nonlinear shallow water waves.