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

Consider the wave equation in one dimension for a function u(x,t)

\frac{d^2u}{dt^2}=\alpha^2 \frac{d^2u}{dx^2}

with the discretization ( \Delta t=1 and \Delta x=1)

u_i(t+1) = \alpha^2 [u_{i+1}(t)+u_{i-1}(t)] + 2u_{i}(t)[1-\alpha^2]-u_i(t-1)

Now the "technical" question. It starts by adding, to the right hand side of the discrete version, a function v_i(t-1):

u_i(t+1) = \alpha^2 [u_{i+1}(t)+u_{i-1}(t)] + 2u_{i}(t)[1-\alpha^2]-u_i(t-1)\mathbf{+v_i(t-1)}

Such that when I choose v_i(t-1) \equiv u_i(t-1) I cancel the terms with time dependence t-1. 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!