Linear Shallow Water Equations

This Chapter introduces the reader to linear hyperbolic systems derived from the non-linear shallow water equations and to basic notions on wave propagation. The purpose is two fold. First to interpret the physical meaning of linearised models and second, practice on the application of salient mathematical notions on hyperbolic partial differential equations, such as eigenvalues, eigenvectors, hyperbolicity and the construction of analytical solutions to the general initial-value problem and its special case, the Riemann problem. The final section is a case study based on a more physically oriented, alternative approach to linearisation, and on which basic mathematical concepts on hyperbolic equations are consolidated. All contents are particularly useful in a teaching/self-studying setting. Useful background is found in Chaps.  1 and 2 .