Damped Wave Equation

This Mathlet illustrates the solution to the wave equation representing a damped plucked string. Modeled by the partial differential equation \(u_{tt}(x,t) + bu_t(x,t) = u_{xx}(x,t)\)

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

Toggles below the large graphing window control what else is displayed.

["Solution"] shows the solution as a red curve. Solution is in quotes because we actually plot the Fourier sum using a large number of terms. This is visually exact unless the plucked point is near the edge of the window, in which case Gibb's phenomenom becomes apparent.
[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.

The damping slider controls the amount of damping. The mathlet allows for the damping coefficient to be negative.

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 \(0 \leq x \leq 1\)).