The motion of a single Fourier mode of the plucked string is an example of transient, free decay of linear, coupled, damped oscillators. It shares the rarely discussed features of the generic case, e.g., possessing a complete set of non-orthogonal eigenvectors and no normal modes, but it can be analyzed and solved analytically by hand in an approximation that is appropriate to musical instruments' plucked strings.

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Assuming damping proportional to velocities allows the use of all the analytical machinery of linear systems. It is a reasonable approximation for good musical strings because it is the transduction of string energy into sound energy that is the dominant string damping. So most of the power lost by the string goes into the power of the sound. The string energy lost in a single cycle is proportional to the amplitude-squared (and, hence, average velocity-squared) in that cycle, just as in viscous damping. The details within a cycle may be more complex because the mechanism is actually a coupling to another linear system.

18.

The analysis of coupled, damped strings in Ref. 6 likewise uses near degeneracy as a key approximation. However, the methods and tools of analysis used there assume considerably greater sophistication on the part of the reader. Also, with somewhat different goals, several of the very general features are not remarked upon as such.

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