Dynamics of the solutions of the water hammer equations

A water hammer is a pressure wave that occurs, accidentally or intentionally, in a filled liquid pipeline when a tap is suddenly closed, or a pump starts or stops, or when a valve closes or opens. A water hammer wave propagates through pipes reflecting on features and boundaries. This phenomenon is governed by a pair of coupled quasi-linear partial differential equations of first order, that are usually solved using the method of characteristics.In this note we provide a representation of the solution using an operator theoretical approach based on the theory of C0C0-semigroups and cosine operator functions, when considering this phenomenon on a compressible fluid along an infinite pipe. We provide an integro-differential equation that represents this phenomenon and it only involves the discharge. In addition, the representation of the solution in terms of a specific C0C0-semigroup lets us show that hypercyclicity and the topologically mixing property can occur when considering this phenomenon on certain weighted spaces of integrable and continuous functions on the real line.