In this paper, within the framework of the Method of Differential Constraints, the celebrated p-system is studied. All the possible constraints compatible with the original governing system are classified. In solving the compatibility conditions between the original governing system and the appended differential constraint, several model laws for the pressure p(v) are characterised. Therefore, the analysis developed in the paper has been carried out in the case of physical interest where p=p0v−γ, and an exact solution that generalises simple waves is determined. This allows us to study and to solve a class of generalised Riemann problems (GRP). In particular, we proved that the solution of the GRP can be discussed in the (p,v) plane through rarefaction-like curves and shock curves. Finally, we studied a Riemann problem with structure and we proved the existence of a critical time after which a GRP is solved in terms of non-constant states separated by a shock wave and a rarefaction-like wave.

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