The wave vector is a vector which points in the direction of propagation of the wave. It has the units of cm-1 and defines the vector in reciprocal space (HERE SHOULD BE THE LINK TO VLAD'S PROJECT). In contrast to the electromagnetic waves the wave vector, k, of lattice vibrations has discrete spectrum within defined limits.
Lets consider the traveling wave:
where 
is a
wave vector.
The possible values of k depend on the boundary conditions. For periodic boundary conditions the only requirement is that U(x)=U(x+L), where L is the length of the chain which gives the values of wave vector:
Now we can see that dispersion relation is not strictly speaking a continuous curve but rather a series of closely spaced points representing the possible modes of vibrations. Figure below is the dispersion relation for diatomic linear lattice.
The spacing between the dots is equal to 
where N is the number of atoms in the lattice.