When numerical methods are applied to the computation of stationary waves, it is observed that "numerical waves" are dispersive for high wave numbers. The numerical wave shows a phase velocity which depends on the wave number "k" of the Helmholtz equation. Recent works on goal-oriented error estimation techniques with respect to socalled quantities of interest or output functionals are developing. Thus, taken into account such aspects, the main purpose of this paper is a posteriori error estimation through of a assessment of the numerical wave number in finite element solution fot he simulation of acoustic wave propagation problems adressed by Helmholtz equation. A method to measure the dispersion on classical Galerkin FEM is presented. In this analysis, the phase difference between the exact and numerical solutions is researched. Fundamental results from a priori error estimation for one-dimensional are presented and issues dealing with pollution error at high wave numbers also are discussed.