The von Neumann analysis is carried out to study the dissipation, dispersion and stability limits of the unsteady linear wave equation solved by the standard 4-stage Runge-Kutta method with several widely used spacial differencing schemes, including 2nd order central differencing, 2nd order upwind, 3rd order and 4th order biased upwind, and 4th order central differencing. The 2nd order Lax-Wendroff scheme and the 2-stage Runge-Kutta method are also analyzed as references. For a central differencing with the 4-stage Runge-Kutta method, there is a CFL limit, under which the solution is dissipation free. The dissipation free CFL limit is far below the stability CFL limit. There is also a CFL limit under which the dispersion error of a central differencing scheme is independent of CFL number. The dispersion error exists for all the schemes studied. The numerical results indicated that the dissipation and dispersion error of upwind schemes with 4-stage Runge-Kutta method are independent of the CFL number under the CFL stability limit. For the wave equation with a low frequency solution studied in this paper, the 4th order central differencing and the 4th order biased upwind differencing have similar level of accuracy.