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For example, a simple wave in the form $$y(x,t)=A_{0}sin(kx-wt)$$ will never disperse no matter what medium it's in because there are no "groups" to have a group velocity, right?

As I understand it, to have any dispersion you need a wave in the form $$y(x,t)=A_{0}sin(k_{1}x-w_{1}t)+A_{0}sin(k_{2}x-w_{2}t)=2A_{0}sin(k_{3}x-w_{3}t)cos(k_{4}x-w_{4}t)$$

But this is just the interference of two waves, so can you only have dispersion when you have more than one wave (of different frequency) interfering? So do pulses disperse because, looking at it from a Fourier analysis perspective, they are built from a bunch of waves of different frequencies?

thanks