Why does different wavelengths of light travel at different speeds in an optical fibre core? I've recently read about material dispersion in an optical fibre which is caused when light that enters have several different wavelengths. What I don't understand is why would wavelength affect the speed of light in the core in anyway?
 A: This is because the speed of light inside the fibre depends on the index of refraction $n$:
$$
v=\frac{c}{n}\, .
$$
In turn, $n$ depends on the wavelength.  It is possible to compute this dependence based on the response of the material to the electric field of the wave, which leads to the permittivity $\epsilon$ to depend on the frequency $\omega$ of the wave, and thus on $\lambda$.  It is simplest to observe this experimentally.
A: In multimode fibre there are a number of different modes each of which has a different $\omega(k)$ relation: Typcally we will have something like 
$$
\omega(k)=\left(\frac{c}{n_{\rm glass}}\right)\sqrt{k^2+\Omega_{mn}^2} 
$$
in which the frequency has a lower cutoff $\Omega_{mn}$ below which thelight cannot propagate in the fibre. The numbers $m$, $n$ label the different modes which look like standing waves in the transverse direction and propagating waves in the direction along the fibre.
This is quite different from dispersion in bulk glass where it is the refractive index $n_{\rm glass}$ which varies with frequency.  The different modes can be loosely though of as the light bouncing (by total internal refrcation) off the cladding of the fibre at different angles. If the light bounces at a lorger angle to the directon of propagation. it spends moretime going sideways and less time going in the direction along the fibre --- think of a sailboat tacking at an angle to the direction in which it wants to go -- and so makes its way along the fibre at a slower rate.   
The wikipedia article https://en.wikipedia.org/wiki/Optical_fiber#Multi-mode_fiber
has some illustrations of this. 
A: When light enters a medium such as glass or water it slows down. If it enters at an angle the slowing causes the path of the light to bend, which is why a lens works. The colour (equivalent to wavelength) affects how much it slows down, and therefore how much it bends, which is why a prism produces colours and why there are colours in a rainbow. 
The reason for this slowing has to do with the way the light waves (which are actually variations in the electric and magnetic fields) are affected by the (electrically charged) electrons in the medium as the waves pass through the atoms. Different wavelengths of light will have slightly different interactions with the electrons.
To complicate things, the speed of the light is only indirectly related to the time signals take to travel through an optic fibre. When light within a piece of glass strikes the inside surface at certain angles it may be unable to leave the glass, instead being totally internally reflected in the glass. Optic fibres take advantage of this phenomenon to stop the light from escaping the fibre (as it otherwise would). The angles involved depend on the colour/wavelength of the light. If you change the angles you change the distance the light must travel, and so you change the speed of the signal in the fibre
