Phase change of transverse waves

Question

My textbook states that a transverse wave on a string, which has a fixed end, is reflected at the fixed end and the phase is shifted by π. No further explanation is given. Here I have plotted a wave (blue) given by $y=\sin \left( t-x \right)$ . So the reflected wave should have the equation $y=\sin \left( t+x+\pi \right)$, which is plotted in red. The resultant wave after reflection is the green plot. From what I understand, if the clamped end was at, say, $x=5$, the phase change couldn't have been π, since there would need to be a node at the clamp. So is the phase change different for different positions of the clamp? If is use trigonometric addition and set the amplitude of the resultant wave equal to 0 at $x=5$ in the above case, I get $\pi-10$ as the phase change of the reflected wave. Is this correct?

Answer 1

So the reflected wave should have the equation $y=\sin \left( t+x+\pi \right)$

To get this formula for the reflected wave, you assumed the reflection happens at $x=0$.

If you want to have a reflection at $x=5$, you need to shift the origin before inverting the spatial parameter.

So your incoming wave is $y = \sin(t-(x-5)-5)$, and your reflected wave is $y_r = \sin(t+(x-5)-5+\pi)$ or $y_r = \sin(t+x -10+\pi)$.

Here's the plot showing the situation at $t=0$:

Blue: incoming wave. Green: reflected wave. Red: Standing wave (superposition of incoming and reflected waves).

This shows there is indeed a node in the standing wave pattern at $x=5$.