This paper deals with the numerical resolution of a linear wave system using the Godunov scheme with a centered discretization of the pressure gradient. The interest in such schemes is motivated by the low Mach number accuracy problem. We have shown that for both steady and unsteady flows, an oscillatory mode appears in the numerical solution. This can be explained by the loss of the Total Variation Diminishing property on the characteristic variables. Moreover, we have illustrated numerically that the long time numerical solution does not converge to the expected steady state. In addition, the existence of the oscillatory mode in the numerical solution jeopardizes the mesh convergence rate of the scheme.