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The Phase difference of 2 travelling waves is how much one wave has shifted from the other, 'angle wise'

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It's constant if both waves travel at the same speed. https://www.desmos.com/calculator/lrekw2jxpd

I learnt that the phase difference of a reflected wave is $\pi$, but what does 'phase difference' even mean here. The waves are travelling in opposite directions and hence there's no 'phase difference' (or at least a constant one) https://www.desmos.com/calculator/umtrrvxlkn .
If you reflect a wave, you can always tell a time, t for which the phase difference between the original and reflected wave is $0, \pi, 2\pi..$(any real number). So what do people mean when they say that the phase difference is $\pi$?

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It means that when primary wave is expressed as $$ y_p(x) = A_p\cos (\Omega t - kz),~~~A_p > 0 $$ so it moves in direction of $z$-axis, and reflecting wall is at $z=0$, then reflected wave is expressed as $$ y_r(x) = A_r\cos (\Omega t + kz +\pi),~~~A_r > 0 $$ which means that at $z=0$, the reflected wave value $y_r(0)$ has opposite sign to incoming wave $y_p(0)$.

So the phase shift $\pi$ refers to values of both waves on the boundary.

This shift $\pi$ happens only for the reflected wave that is reflected back to medium with lower phase speed. For example, when light travelling in air is reflecting from the air-glass boundary back to air.

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  • $\begingroup$ So the 'phase difference' is ONLY about z = 0 (the boundary) ,where the waves' corresponding points are ALWAYS pi apart right? $\endgroup$ Feb 7, 2021 at 15:18
  • $\begingroup$ Yes the phase difference concers only the point where the reflection happens. The phase is off by $\pi$ only some reflections, see above. $\endgroup$ Feb 7, 2021 at 16:23

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