Elementary Waves in Shallow Water

This Chapter is devoted to the study of elementary waves emerging from the solution of the Riemann problem for the augmented one-dimensional and for the split two-dimensional shallow water equations. The dam-break problem is introduced as a physical, motivating example of a special case of a Riemann problem. Four possible wave patters in the solution of the Riemann problem are identified, each comprising rarefactions, shocks and Contact discontinuitycontact discontinuities or Shear wave shear waves. For each wave type, mathematical relations across the wave structure are established. These include generalised Riemann invariants and Rankine-Hugoniot jump conditions. Useful shock relations are established for left and right shocks, connecting shock speed with shock Froude number. These relations can be readily used to setup shock test problems with exact solutions, to test numerical methods. Finally we study shock waves from conservative formulations of the shallow water equations in terms of physical variables, rather than conserved variables. It is found that such shocks are slower than those from the conserved formulation using the conserved variables. The Chapter is concluded with a list of suggested exercises. Useful background is found in Chaps. 1 , 2 and 4 . The contents of this Chapter will be used to solve the complete Riemann problem exactly in Chaps. 6 and 7 .