Graduate Are superpositions of waves with different c still waves?

[greypilgrim]

Hi.

As far as I know, superpositions of waves are normally considered to be waves too, even in dispersive media. But how can they still be solutions of a wave equation of the form
$$\left(\frac{1}{c^2}\frac{\partial^2}{\partial t^2}-\Delta\right)u=0$$
if $c$ isn't the same for all of them (dispersion)?

 

[DariusP]

I think a superposition of waves is called a complex wave

 

[Delta2]

Superposition of monochromatic waves with different phase velocities for example two monochromatic waves $u_1,u_2$, with phase velocities $c_1,c_2$ is indeed considered to be a wave, however the superposition wave $u=u_1+u_2$ does not satisfy that wave equation you wrote for any $c$, though each of $u_1,u_2$ satisfies that wave equation for $c=c_1$ and $c=c_2$.

What happens in dispersion is that the wave's envelope is distorted as it travels through the medium, and though we can say that the superposition wave travels with the group velocity $v_g$ , the superposition wave does not satisfy that wave equation neither for $c=v_g$.