The phase difference is related to path difference delta x by
Answers
path difference is the difference in path traversed by the two waves , measured in terms of wavelength of the associated wave. It has a direct relation with phase difference. Phase difference decides the nature of interference pattern but phase difference is found out by path difference. Phase difference is related to quantum mechanics. If path difference b/w 2 waves is integral multiple of wavelength, which satisfies condition for constructive interference. Whereas, if path difference b/w 2 waves is odd multiple of half wavelength, it satisfies condition for destructive interference.
whereas phase difference is the difference between some reference point in 2 waves. Its how much one wave is shifted from the other. For eg. at the origin, if one wave's displacement is zero and the other one has some displacement, then there is a shift, which is their phase difference. For eg. sine wave is zero at the origin, but cos wave is not zero at the origin, its zero at pi/2. So the phase difference is pi/2. Infact, cos wave is just the sine wave phase-shifted.
If two waves have zero phase difference, then their crests occur at the same time and so do their troughs. Its like moving together. They will add up (constructive interference).
For only one wave, phase difference means how much the wave is shifted from the origin. Like in your example, Acos(wt + Φ), at time t=0, cos has some value other than 1. So, the cos wave is slightly shifted by Φ. Physically, what this means is when u started measuring the time, the system oscillates, not from either origin or amplitude, but somewhere in between.
Ofcourse, the concept of phase difference is not useful if there is only one wave. You could always choose time to start from when the system is at an extreme position. Phase difference concept is useful when there are atleast 2 waves, then when u start the time when one wave starts from zero, the other wave could already be in some other position.