Unsteady flow surges in rivers can be idealized as a transition between lower steady flow downstream and higher steady flow upstream. Simple wave and monoclinal wave models are analytical solutions of the dynamic wave equations that can be used to model these surges. The simple wave model neglects bed slope and flow resistance, simulating surge propagation over relatively short distances. The monoclinal wave profile evolves progressively, and this model is generally applicable at larger distances. These models are easy to use and generally apply to different flow conditions. However, both models lack criteria to determine their suitability for specific applications. The objectives of this paper are to develop and to assess criteria to guide simple wave and monoclinal wave model applications and to use the models to gain insights into the inertia and slope--flow resistance contributions to momentum and the nonlinear effects of increasing surge amplitude. Simple wave model application criteria include a dimensionless time, the amplitude to initial amplitude ratio for a linear dynamic wave traveling downstream, a distance scale, and the ratio between the measured surge celerity and the c+ dynamic wave celerity of the high flow upstream. The length of the monoclinal-diffusion wave profile gives the minimum distance needed to develop a steady monoclinal wave profile. Model capabilities to describe flow surges and the adequacy of the application criteria are evaluated with data from laboratory and field studies covering a wide range of conditions. Profile celerity and high flow velocity behind the surge are always greater for the simple wave than for the monoclinal wave because of the absence of slope and flow resistance effects. The differences between the models are enhanced as Froude number decreases and as surge amplitude increases.