Harmonics and Fourier analysis

A full period (T = 2π/ω) of an oscillation is shown below, as well as the amplitudes A_k = 2|c_k| = 2|c_-k| and phases φ_k = arg c_k = -arg c_-k of the corresponding Fourier series:

x(t) = ∑ c_n e^inωt = ∑ A_k cos(kωt+φ_k), (n = ...-1,0,1,...; k = 0,1,...).

One can choose below the oscillation profile and the number of harmonics: N. The sum of the latter is displayed in blue.
From there one can change amplitudes and phases by moving the sliders with the mouse.
If Full series? is selected, the sum of the infinite series is displayed in red. This graph does not change when you change amplitudes or phases, for it corresponds to the exact selected profile.
To get information on an element, put over it the mouse pointer to see the corresponding tooltip.

Activities

Select different oscillation profiles and discuss what happens when an amplitude or a phase is changed.

Use Linear or Square profiles to analyze the Gibbs phenomenon for different values of N: the convergence is clearly slower near the discontinuity points.

This is an English translation of the Basque original for a course on mechanics, oscillations and waves.
It requires Java 1.5 or newer and was created by Juan M. Aguirregabiria with Easy Java Simulations (Ejs) by Francisco Esquembre. I thank Wolfgang Christian and Francisco Esquembre for their help.