Nonlinear evolution equations for two-dimensional surface waves in a fluid of finite depth

Two-dimensional weakly nonlinear surface gravity-capillary waves in an ideal fluid of finite water depth are considered and nonlinear evolution equations which are correct up to the third order of wave steepness are derived including the applied pressure on the free surface. Since no assumptions are made on the length scales, the equations can be applied to a fluid of arbitrary depth and to disturbances with arbitrary wavelength. For one-dimensional gravity waves, these evolution equations are reduced to those derived by Matsuno (1992). Most of the known equations for surface waves are recovered from the new set of equations as special cases. It is shown that one set of equations has a Hamiltonian formulation and conserves mass, momentum and energy. The analysis for irrotational flow is extended to two-dimensional uniform shear flow.