The method used to separate surface and internal tides ultimately defines properties such as internal-tide generation and the depth structure of internal-tide energy flux. Here, we provide a detailed analysis of several surface-/internal-tide decompositions over arbitrary topography. In all decompositions, surface-tide velocity is expressed as the depth average of total velocity. Analysis indicates that surface-tide pressure is best expressed as the depth average of total pressure plus a new depth-dependent profile of pressure, which is due to isopycnal heaving by movement of the free surface. Internal-tide velocity and pressure are defined as total variables minus the surface-tide components. Corresponding surface-and internal-tide energy equations are derived that contain energy conversion solely through topographic internal-tide generation. The depth structure of internal-tide energy flux produced by the new decomposition is unambiguous and differs from that of past decompositions. Numerical simulations over steep topography reveal that the decomposition is self-consistent and physically relevant. Analysis of observations over Kaena Ridge, Hawaii; and the Oregon continental slope indicate O (50 W m-1) error in depth-integrated energy fluxes when internal-tide pressure is computed as the residual of pressure from its depth average. While these errors are small at major internal-tide generation sites, they may be significant where surface tides are larger and depth-integrated fluxes are weaker (e.g., over continental shelves).