Let's look again at the difference scheme we derived for the 1-D wave eq, with the special time/space step :
The velocity sample is a two-dimensional sequence with a time index and a spatial coordinate index.
Suppose we now take the DTFT with respect to the spatial index :
where here is the spatial spectrum of the solution at time , and is the spatial frequency variable. We can also write this in vector form as:
Note that the state of the system is completely determined by and .