Two dimensional square lattice is considered when the forces between the lattice particles are expressed both linearly and quadratically dependent on the spring elongations. The shift operator approach, firstly developed by [1] and applied to the one-dimensional linear problem, is extended on the two-dimensional nonlinear case. The discrete strain energy and the discrete governing nonlinear equations of motion are obtained. Also, the continuum nonlinear equation for the plane longitudinal waves propagation is obtained in a weakly nonlinear case.