Measuring the harmonics of a vibrating string I have a string which is vibrating at a fixed frequency, say 1 kHz.  I know it is vibrating in a purely first harmonic mode because of the (sinusoidal) drive function I am using.  With a microphone I measure the sound wave and do a frequency analysis of this.  I measure several harmonics in this spectrum.  Is this possible? Does the environment of the string generate higher harmonics?
 A: A linear system driven by pure sine wave will produce a response only at the fundamental frequency.  There will be no harmonics or overtones.  Harmonics are multiples of the fundamental frequency. Overtones are higher order resonant modes of the system. For a vibrating string, the overtones are very close in frequency to multiples of the fundamental frequency. 
In order for harmonics to be present, either the source is not a perfect sine wave or there is some nonlinearity in the system. An even-order nonlinearity (e.g. quadratic) will create only even harmonics. An odd-order nonlinearity (e.g. cubic) will create only odd harmonics. The nonlinearity could be, for example, that Hooke's Law is not obeyed perfectly.  
In order for overtones to be excited, there must be some mode-coupling present so that energy can flow from the fundamental mode into higher-order modes. This can occur in any real system with damping and especially in situations where energy is strongly focused in a small region (e.g. bending at the ends of the string).
