On noninertia wave versus diffusion wave in flood routing

This note intends to address a misnomer in flood routing. The noninertia wave, which ignores both local and convective inertia terms in the momentum equation, is a simplification to the full dynamic wave. While the diffusion wave refers more generally to the wave whose induced disturbance in flow is analogous to the diffusion of particles or heat. For the purpose of clarification, these waves are mathematically formulated and physically interpreted. It is demonstrated that the diffusion wave can be mathematically formulated from different levels of shallow water wave approximations with the assumption that the celerity and coefficient of hydraulic diffusivity are step-wise constants. Both the linear and nonlinear perspectives of waves are discussed. The noninertia wave is demonstrated to be one of the special cases of the diffusion wave.