if u have a fretted note the string vibrates between the fret and the bridge. but this vibration is not simple. it is made of many sets of waves which combine to make the sound of a guitar string. ever wondered why a note on a cello or piano etc sounds different. well its cos there are different waves within the wave that is the vibration of the string. the wavelength of each component vibration is the length of the vibrating string divided by an integer. ( whole number)
so the constituent waves have wavelengths the length of the string, half the length of the string a third of the length of the string. and these wavelengths correspond to different frequencies by an inverse relationship ( half the wavelenght double the frequncy). each wavelength vibrates different amounts, this gives different instruments there diferent tone.
in an octave all the frequencies (different wavelengths) in the higher of the two notes are present in that of the lower octave.
as the wavelength of the higher note is corresponds to the wavelength thats is half of the lower one etc and all the other ones match up.
so basicaly this is why octaves match so much cos all the sounds of the higher note is contained in that of the lower.
i realise this may go way over your head and it doesnt really matter but i felt someone might be interested.