Numerical methods for the compressible Euler equations may encounter difficulties with the zero Mach number limit. For instance, it is well-known that the Roe scheme lacks accuracy when dealing with low Mach number flows, especially on Cartesian grids \cite{guillard1999behaviour}. In response to this issue, numerous modifications to the Roe scheme have been proposed. Many of these modifications fall into the category of Roe-pressure-centered schemes \cite{rieper2011low,Dellacherie2016_all_Mach}. Unfortunately, these schemes introduce other issues, such as the appearance of oscillatory modes in the numerical solution \cite{Dellacherie2009_checkerboard}. Leaning on an asymptotic analysis of the Euler equations as the Mach number goes to zero leads to a link with the long-time behavior of a linear wave system. Similarly, at the discrete level, Roe-pressure-centered schemes are known to be asymptotic consistent with a discretization of a linear wave system, in which a centered discretization of the pressure gradient is used. This discretization is referred to as the pressure-centered scheme. Due to the mentioned link, the issue of spurious modes arises also in the solution of the pressure-centered scheme, as illustrated in \autoref{fig:Ondes:2D:iso-contours numerical sol:pressure-centered-scheme:velocity field}. In this talk, we focus on the pressure-centered scheme, we identify the origin of the oscillatory modes problem and characterize these spurious modes. Furthermore, numerical simulations reveal that these modes significantly compromise the convergence of the numerical solution towards the expected one.