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

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

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

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

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

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

$\displaystyle 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 $\displaystyle v_i(t-1) \equiv u_i(t-1)$ I cancel the terms with time dependence $\displaystyle t-1$. 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!