Damping of not harmonic waves You pluck a (guitar) string so that you create a wave with harmonic frequency and a wave with not-harmonic frequency.
Which one will be heard longer? Why?
Or ask it differently: Is the wave with not-harmonic frequency damped a lot faster than the harmonic one?
 A: In a typical treatment of an ideal (under)damped harmonic oscillator, the damping term is proportional to velocity, but is otherwise assumed to be independent of frequency.
In this case the rate at which the oscillations diminish goes as $\exp (-\gamma t)$, where $\gamma$ is some sort of damping constant.
Of course what must be true is that the amplitude of the anharmonics will presumably be lower than the fundamental and harmonics, so presumably if they all decay at the same rate, the anharmonics drop below the threshold of hearing more quickly than the harmonics. 
However the last comment you have made makes me think you want to know the answer to a different question, which is why the harmonics have higher amplitudes to begin with? The answer to this is that if you stimulate the string across a broad range of frequencies, the response of the string is much stronger at frequencies corresponding to the fundamental mode of vibration and its harmonics. This is the basic phenomenon of resonance.
