How do the particles in a standing wave vibrate in phase within half a wavelength? I would like to have a better understanding of phase difference when talking about stationary waves. How exactly are particles in phase within half a wavelength in a stationary wave?
 A: A standing wave is formed when two identical waves travelling in opposite directions meet and superimpose on each other. This usually happens when one wave is the reflection of the other.
Now, to better understand this, see this animation, 

It is pretty apparent that the black particle that doesn't appear to be moving will be in phase with the particle that is to the right of the particle that is going up and down. That is, it will be in phase with the particle after half a wavelength (after one crest/trough).
Similarly, the particle that is going up and down will be in phase with the particle that is on the next crest/trough. That is, after half a wavelength.
A: In phase simply means that the particle are moving at the same direction at all times. For a standing wave, this is achieved only if they have the same amplitude at all times.
Using parthvader's animation below, if you take any of the two black dots, that represent a particle, you can see that all the points that move in parallel to that point are all half a wavelength from each other.
