Using complex numbers to describe oscillations

Generally, oscillations are described mathematically using differential equations. Up until this point in the discussion of oscillations, numerical differential equation solvers have been used and the formulas for the oscillation frequency and resonance frequency have just been stated. Next some analytic solutions will be derived. The analytic solutions typically make use of complex numbers so first Euler's formula $e^{i\omega t} = \cos(\omega t) + i\sin(\omega t)$ will be discussed.

The red ball represents the position of the complex number e^iωt as it moves through the complex plane. Imaginary numbers are plotted verically and real numbers are plotted horizontally. The blue ball represents the position of cos(ωt) and the green ball represents the position of isin(ωt). The simulation on the left is a graphical representation of the formula on the right.

Oscillations that can be described by sin(ωt) or cos(ωt) are called harmonic oscillations. From the simulation it is clear that there is a relationship between circular motion and harmonic oscillations. If you look at the motion of the red ball from above, it moves in a circle but if you look at the motion of the red ball from the side, it executes harmonic motion.

The relationship between circular motion and harmonic oscillations is described easily using complex numbers. Sometimes when we observe a harmonic oscillation it is convenient to imagine that we are looking at circular motion from the side. We can't measure the component of the motion in the imaginary direction, we just imagine it.