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 th normal mode is given by
2) The frequencies are given by
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.
Write initial-condition code to excite your string in any given normal mode with any given amplitude. Using a tension sufficient to stretch your string by at least twenty percent of its original length, try different normal modes at various amplitudes (ranging from “barely visible” to “large”). Does your model reproduce the expected qualitative behavior of the ideal vibrating string? In which regimes?
How does the behavior depend on the number of masses you have chosen? Should this affect the behavior at all? Is using a large more critical for modeling the low- normal modes, or the high ones? Why?
Now, write initial-condition code to excite your string with a Gaussian bump with any given width and location. Excite your string with narrow and wide Gaussians of modest amplitude, located both near the endpoints and near the center of the string. By observing the animation, comment on whether you have excited mostly the lower normal modes, or the higher ones.
If you have access to a stringed instrument (guitar, violin, ukelele, etc.), excite your real string in these ways (by plucking it with both your thumb and the edge of your fingernail, both near the center of the string and near the endpoint). How does the timbre of the sound produced correspond to the answers to your previous question?