A general formulation for surface deformation generated by a initial localized free surface disturbance is established in double integral form. The initial condition characterizing the localized disturbances can be 1) a movement of the free surface characterized by a velocity distribution, 2) a static departure of the free surface from the still water level, or 3) an impulse on the free surface. Solutions are obtained for a number of cases in which the initial disturbances are mathematically defined, allowing a first integration by Hankel transform. Closed form solutions are obtained in the nondispersive mode, while numerical solutions are obtained in the general cases. It is found that the initial disturbances result in nonoscillatory movements at the origin. Also, the wave motions are little affected by dispersion. The oscillations which are known to exist at a distance are only the results of linear dispersion in deep and intermediate water depth. Also the vertical rebound following the collapse of cavities (in the form of a spike, jet or drop) is explained by discontinuities in the original perturbations.