Part 4: The low-amplitude limit, qualitatively

If the vibrational amplitude is sufficiently small, the trigonometric functions involved in the geometry here can be replaced with the first nontrivial term in their Taylor series. (These are often called the “small-angle approximations.) When this is done, the system becomes perfectly linear and can be shown to obey the classical wave equation. This leads to the behavior you are probably familiar with, characterized by three main properties:

1) The “normal modes”, oscillatory patterns of definite frequency, are sine waves with a node at either end (and possibly other nodes in the middle). Specifically, the nth normal mode is given by

(6)y(x,t)=AsinnπxLcosωnt,
where A is some arbitrary amplitude.

2) The frequencies ωn are given by

(7)ωn=πvnL
where v is the wavespeed, given by
(8)v=T/μ.
(Recall that primed quantities correspond to their values after the string has been stretched.)

3) Since the system (in the ideal, low-amplitude case! – which is all you can solve easily without the computer) is perfecty linear, any number of normal modes, with any amplitudes, can coexist on the string without interfering. Any arbitrary excitation – like a plucked guitar string – is a superposition of many different normal modes with different amplitudes.

Let’s now see how well your string model reproduces these ideal properties.