Periodic Surface Waves Produced by Non-Periodic Disturbances

[Caveman11]

This is part of a past paper I am trying to work through before a physics of fluids exam in a month.The angular frequency ω of a periodic surface wave with wavenumber k on
deep water is
ω = sqrt(gk)
where g is the gravitational acceleration.
Obtain an expression for the wave’s phase velocity in terms of its wavelength
and explain how a localized non-periodic disturbance far out at sea can lead
to approximately periodic surface waves at the shore.
Relevant equations:

$$V_{p}$$=$$\frac{\omega}{k}$$ (Phase Velocity)
3. The Attempt at a Solution : I obtained a value for $$V_{p}$$ as:

$$V_{p}$$=$$\sqrt{\frac{g\lambda}{2\pi}}$$

Which I assume to be correct. However I can't explain the fact that you can get periodic waves from non-periodic motion. The only explanation I could think of was that the waves with the same wavelength travel together as they have the same phase velocity where as longer wavelength waves have a higher phase velocity and can overtake the slower ones. This would lead to after a while all the different periods being grouped together?Many Thanks

 

[olivermsun]

Yes. That's "dispersion" for ya!

 

[Caveman11]

I didn't expect it to be be that, Thanks