discretization of wave equation ("technical")

Hi,

I have a question regarding the discretization of the wave equation.

Consider the wave equation in one dimension for a function

with the discretization ( and )

Now the "technical" question. It starts by adding, to the right hand side of the discrete version, a function :

Such that when I choose I cancel the terms with time dependence . Doing that modifies the discretization of the 2nd time derivative, and the resulting discrete equation is not anymore a "genuine" wave equation. *But then, how would the differential form of the last equation look like? *

Any help would be highly appreciated!! Thanxxxxxxxxx!