Can there be traveling waves on a string that is attached at both ends (like a guitar string), or does such a configuration allow only standing waves ?
Of course there can be (and are) traveling waves on a string - when you pluck anywhere except at the center, you are generating an asymmetrical impulse that will travel up and down the string. The fact is that your "pluck" consists of many different frequencies, and as these travel back and forth two things happen:
- the highest frequencies will be damped most quickly due to dissipation (to the air, the supports, and internal friction in the string)
- waves that do not "fit" tend to cancel after a few round trips (interference) leaving just the standing wave pattern.
It is only the time evolution of the traveling wave that turns it into two waves of equal amplitude and discrete frequencies that travel in opposite directions - and that you recognize as standing waves.
A detailed mathematical analysis can be found in this paper