Take into account the simple case of a wave that satisfy the <PERSON> equation (for simplicity 1-D case): $$ u_{tt} - c_{0}^2 u_{xx} $$ You know that the most general solution of this equation is given by the superposition of two profiles that does not change in form during the motion: $f(x-ct)$ that traslates towards positive $x$ and $f(x+ct)$ that traslates towards negative $x$. The general solution reads: $$ u(x,t) = f(x-c_{0}t) + f(x+c_{0}t) $$ It is the so-called "stationary wave", that is a superposition of two profiles with the same frequency, that does not change during the evolution according to the <PERSON> equation. Now let's get to the point: the D'Alambert equation above is a linear (superposition of solution is a solution yet) and NON DISPERSIVE equation, the fact that is not dispersive, practically means that you can change the frequency (or the wave-number) of your traslating profiles, without changing the phase velocity of propagation, that in any case remains $$c= \frac{\omega}{k}=\pm c_{0}$$. You can see that in this first case the dependece $\omega(k)$ is linear and as a consequence of this fact, you will have that your traslating profile will not change in time, as written above. This also can be thought in terms of Fourier components: just think about that in this way you can describe a stationary wave like a sum over components of different wave-number. 795
In other words, the most general way to write the solution can be by developing a Fourier series, incuding all the possibile wave- vectors $k$ in your solution, without losing of generality. For example, a non dispersive wave is the electromagnetic one in the vacuum.
On the other hand, a DISPERSIVE wave-equation, is an equation in which the dispersion relation is something like $$c= \frac{\omega(k)}{k}$$. Where the dependece $\omega(k)$ is not linear in this second case. This cleary imply that the phase velocity of your wave will change, if you modify the frequency (beacause $\omega$ depends on $k$) and so, in this case you cannot have that waves with different frequencies (or wave-numbers) propagate with the same phase velocity. As examples of dispersive wave, think about surface waves on deep water, or electromagnetic waves in a medium. 795