Wave Equation

This Mathlet illustrates the solution to the wave equation representing a frictionless plucked string.

A large graphing window displays the initial condition -- the string of the string at \(t = 0\) -- with a thin yellow polygon, and the plucked point with a diamond.

Toggles below the large graphing window control what else is displayed. Selecting [Solution] shows the solution as a yellow polygon. [Harmonics] toggles display of some of the Fourier components of the solution, in white. A slider at bottom selects the number of harmonics. [Fourier sum] toggles display, in cyan, of the sum of the selected number of harmonics. [Left wave] creates the left-moving traveling wave in the d'Alembert solution, in green, and [Right wave] creates the right-moving traveling wave, in orange.

Grabbing the handle on the time slider selects the time. The display can also be animated using the [>>] key at left. Once the animation has started, the key is replaced by [||], and selecting it stops the animation. The speed of the animation is controlled by a slider marked [Animation speed].

Rolling over the graphing window produces crosshairs and a readout of the values of \(x\) and \(u\). Depressing the mousekey suppresses the crosshairs and repositions the pluck point to the nearest possible point (since it is constrained by \(0 \leq x \leq 1\)).