For two-dimensional unsteady flow in shallow water, real characteristic surfaces exist. Their envelope is a quadric conoid, whose curvilinear rays serve as paths for the transmission of information in t-x-y space. The original equations of flow can be reduced along these rays, called bicharacteristics to equations containing differentiation in one less direction. Simplified numerical methods can be developed, utilizing this property of the equations, that are superior to both finite difference and finite element techniques. Three numerical characteristic networks of varying complexity are tested for accuracy and stability with flow problems that involve both subcritical and supercritical flow. The solution proceeds in specified time intervals by extending the bicharacteristics backwards until they intersect a plane on which the flow conditions can be found by interpolation from known data. Finally, the most accurate numerical network is applied to the computation of a highly two-dimensional wave propagating on a dry bed.