Standing waves - why do wavelengths fit perfectly? When reading about standing waves it is always said that only certain wavelengths are "allowed". I understand that these wavelengths are a requirement for there to be a standing wave due to the boundary conditions, but what does "allowed" mean in this context?
When creating a wave on a fixed string, does this mean that the wavelength will always be an appropriate fraction of the length of the string such that a standing wave exists - i.e it is not possible to create waves with a wavelength that would not create a standing wave on a fixed string?
Or are the wavelengths completely dependent on the source that created the wave, and standing waves are simply the special case/coincidence when the wavelength is appropriate?
 A: It is a good question. The single standing waves are the modes that get emphasized in teaching, but the real motion of strings is due to sums of such modes.
Consider a Slinky. One can make it swing in the fundamental mode or in modes with one or two nodes by adding energy from your hands in just the right intervals. You will feel that, that gives a positive feedback to that mode. That also is what happens at the lip of an organ pipe etc. The standing waves are resonances.
Or one can look at what happens when you make a pulse on the Slinky. Or on this Java simulation by Falstad. The pulse travels back and forth, getting reflected at both ends. So there is a periodic signal, with many harmonics of the fundamental frequency.
Falstad's simulation also has the option of adding a driving force with arbitrary frequency. It is like pushing a swing at arbitrary times: it won't add energy to the system.
A: The reason they are "standing" is precisely because the wavelength fits perfectly. If the wavelength didn't fit, it wouldn't be standing but moving along.
So your thinking is good enough, just the wrong way round.
Why a given piece of string (or, for that matter, a bridge) has a tendency to create standing waves is another issue, but I am certain others would give you a better explanation than I.
