Demonstration: Harmonic Oscillator

Below are the time series and the phase portrait for a harmonic oscillator, subject to a the second-order linear differential equation of the form \[y''(t)+p y'(t)+4 y(t)=0.\]

The oscillator is critically damped when $p=4$. The graphs are depicted in red when the spring is underdamped (or undamped), green when the spring is critically damped, and blue in the overdamped case.

Note below that the critically damped case features one pair of straight-line solutions. That pair splits into two pairs of straight-line solutions when the oscillator becomes overdamped.

You can move the "cross-hair" locator to see a different solution.