I'm asking this question because I did not find any explanation in textbooks.
Consider a string fixed at both ends with an impulse provided to it (like in a guitar). The impulse can be described using Fourier integral and contains all the possible frequencies (with their weights). Nevertheless only the frequencies that satisfy the condition $$f=n \frac{v}{2L}$$ will "survive" and all the other components will disappear in a short time, after some reflections.
I would like to have an approximate, but quite clear, idea of how the wave components not satisfying to $(1)$ will disappear after being reflected some times.
Consider the situation in the picture, which is the case of $n=2$ standing wave (in black) and the red one and green one are the incident and reflected wave.
Because of $(1)$, in this case the incident and reflected wave add up to zero in both ends of the rope at any time $t$ (this is indeed the condition from which $(1)$ is derived).
Anyway, if this does not hold, i.e. the incident and reflected wave do not add up to zero in one of the ends of the rope, what happens to the wave?
In my view these two things must hold anyway:
- The end of the rope is physically fixed so it won't move in any case
- Since the end does not move, the reflected wave is always reflected upside down. So also the components not obeying $(1)$ will be reflected up-side down
But this does not explain why the components not satisfying $(1)$ eventually disappear. And it also looks like point 2. This is quite absurd since, if the waves are reflect upside down (like in picture), but independently from the fact that they satisfy condition $(1)$ or not, it seems to me that they should add up to zero at the ends of the rope in any case.
But this is surely incorrect and I'm quite confused on what happens to the components of the wave not obeying $(1)$, so any explanation is highly appreciated.