Exploring Wave Propagation in a 1D Uniform Lattice

[DoctorDeath64]

If we take a 1D uniform lattice I understand that we can derive a difference equation after using Hookes law and Newtons 2nd law as seen in the section labelled (Scalar wave equation in one space dimension, Derivation of the wave equation, From Hooke's law) in the link below.
http://en.wikipedia.org/wiki/Wave_equation

We assume that the displacement of a mass positioned at point x, is y(x)=exp[i(kx-wt)], therefore in the discrete lattice we obtain the relationship y(n+1)=exp[ika]y(n), where the spacing of between lattice points is a. By substituting the above relationship and the time dependence into the difference equation we obtain a dispersion relation. The difference equations have position varying, how would we introduce a variation of time ?

So I am wondering what does this represent ? Is it that each point on the curve corresponds to a vibration value and a wavenumber, so that when the perfect lattice has a particular wavelength (i.e. wavenumber), all the masses vibrate at a particular frequency and vice versa?

What if I was to introduce a defect ! Then the dispersion relation would no longer hold, so you may get some points in the stop-band. Does this mean if I vibrate the lattice at a localised vibrational frequency that lies in the stop band, that the wavelength will be a specific value and the displacements will decay from the point of the defect ?

A stop-band is a range of wavenumbers at a particular frequency, what does this mean ? Does this mean , if the lattice vibrates at a frequency lying in the stop band that no waves will propagate through the lattice regardless of wavelength ?

Also they say at the edge of the Brillouin zone, standing waves occur where no energy is propagated, therefore the group velocity=0. Surely if no wave propagates through the lattice, the phase velocity is also 0 ? But in the dispersion curve you can select a value k=pi/a and w>0 for the uniform lattice, which is at the edge of the Brilliouin zone, but the phase velocity=w/k which doesn't equal zero ? Therefore the lattice propagates in time ?

Sorry about the long post, I'm just v. confused. Any help would be much appreciated.

 

[BigJDubs]

Yes, each point on the dispersion relation curve corresponds to a particular wavenumber and frequency of vibration. If you introduce a defect into the lattice, the dispersion relation will no longer hold, and some points in the stop-band may appear. This means that if you vibrate the lattice at a localised vibrational frequency that lies in the stop band, the wavelength will be a specific value and the displacements will decay from the point of the defect. A stop-band is a range of wavenumbers at a particular frequency, meaning that if the lattice vibrates at a frequency lying in the stop band, no waves will propagate through the lattice regardless of wavelength. At the edge of the Brillouin zone, standing waves occur where no energy is propagated, and thus the group velocity is 0. However, the phase velocity is not necessarily 0; it can still be non-zero depending on the wavenumber and frequency of vibration. This means that the lattice can still propagate in time, but the energy does not go anywhere.