Investigations of the computational stability of finite difference formulations of the equations governing shallow water, non-linear gravitational oscillations are reported. The investigations consist of empirical analyses of the results of numerical integration of the quasi-two-dimensional and fully two-dimensional versions of the equations. the results obtained in the quasi-two-dimensional experiments suggested the formulation of filters for use in approximating the non-linear terms. By filtering certain high frequency non-linear interactions, it is found possible to archive relatively well-behaved long term integrations. A number of efforts to extend this result to the fully two-dimensional equations are indicated. One procedure was found to be practically successful, but is finally rejected on the basis of the instability of its linear counterpart. The paper concludes a critique of possible extensions of the investigation.