The animation illustrates the principle of Fourier synthesis: Complex periodic waveforms can be generated by superposition of simple sine waves (harmonics). Interactively, the number of harmonics can be varied and different target waveforms can be selected.
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Description of the animation
The animation visualizes the Fourier series, which allows periodic functions to be represented as a sum of sine functions:
[ f(x) = sum_{n=1}^{N} a_n sin(n omega t) ]
In the left area of the animation, so-called epicycles rotate – concentric circles whose radii correspond to the Fourier coefficients. Each circle represents a harmonic of the fundamental frequency. The endpoint of these chained circular motions traces the resulting waveform, which is displayed as a time evolution in the right area.
The Fourier coefficients differ depending on the target waveform:
- Square wave: Only odd harmonics contribute, with coefficients ( a_n = frac{4}{pi n} ). The characteristic overshoots at the edges (Gibbs phenomenon) remain visible even with many harmonics.
- Sawtooth wave: All harmonics are involved, with coefficients ( a_n = frac{2 cdot (-1)^{n+1}}{pi n} ). The alternating sign changes produce the asymmetric shape.
- Triangle wave: Only odd harmonics, with coefficients ( a_n = frac{8 cdot (-1)^{(n-1)/2}}{pi^2 n^2} ). Due to the quadratic decay in the denominator, the series converges faster than for the square wave.
At the bottom of the screen, the individual harmonics are displayed separately, allowing their respective contributions to the overall waveform to be examined.
Interactive controls
The animation offers several possibilities for interaction:
- Number of harmonics (N): The slider controls how many overtones are used for synthesis (1 to 15). As the number increases, the generated curve approaches the ideal target waveform.
- Speed: Adjusts the rotation speed of the epicycles and thus the frequency of the generated wave.
- Waveform: Using radio buttons, you can switch between square, sawtooth, and triangle waves. The semi-transparent white line shows the ideal target shape for comparison.
- Play/Pause: Stops or resumes the animation.
- Reset: Resets the phase and clears the previous waveform.