Math Insight

Applet: The phase of an oscillator

We can represent the state of an oscillator by its phase $\theta(t)$, which lies in the interval $[0,2\pi]$. Since we view the phase as being a periodic variable, we can draw the interval $[0,2\pi]$ as a circle in the complex plane where $\theta$ is the angle between the positive real axis (what you might call the positive $x$-axis) and the vector from the origin to a point on the unit circle. This point is drawn in green, and you can change it by dragging it with the mouse. To display the phase most compactly, we will typically draw the phase of an oscillator as just a point on the unit circle (and omit the vector and angle shown here). The coordinates of the point in the complex plane are given by $e^{i\theta(t)}$ where $i=\sqrt{-1}$.

Applet file: oscillator_phase.ggb