Prove that a travelling wave, at an open boundary is reflected without any phase change.
Verified Answer
In this case, the left end of the string is tied to a ring, which slides up and down without any friction on a rod.
When the pulse arrives at the left end, the ring moves up the rod.
As the ring moves, it pulls on the string, stretching it and producing a reflected pulse with the same sign and amplitude as the incident pulse.
Thus, the incident and reflected pulses reinforce each other, creating the maximum displacement at the end of the string: the maximum displacement of the ring is twice the amplitude of either pulse.
The reflection occurs without any additional phase shift.
In the case of a travelling wave, reflection at an open boundary (such as the open end of an organ pipe) also takes place without any phase change.