We show how diffusive fluxes based on Darcy's law define groundwater hydrographs following sharp perturbations on the hydraulic head. The derived normalized hydrograph is based on a single fitting parameter, the time constant b, and has great semblance to observed discharge hydrographs for many karst springs and small rivers following rainfall events. Observed time constants increase with basin area but at a given site decrease as peak flows increase. The predicted interval between a pulse in head and the resultant peak flow is equal to 2b/3. Longer delays are observed between actual rainfall events and peak flows because the head perturbation must postdate the rainfall.