Welcome back! In this entry in our Demonstration Highlights series, we’re taking a look at Fourier Synthesis. You may recall that we addressed Fourier Analysis in a previous entry, the process of analyzing a waveform by breaking it down into harmonic components. This time, we’re taking the process in reverse. In Fourier Synthesis, we assemble a wave form by adding sine waves together.

 Demo H4-01: The Fourier Synthesizer, with speaker and monitor

Our Fourier Synthesizer demonstration, H4-01 in the demonstration index, lets you generate a sine wave of any frequency between 100Hz and 1,000Hz. The synthesizer then generates harmonics of this frequency, waves with integer multiple frequencies – e.g. 120Hz, 240Hz, 480Hz, etc. You can then choose to add any or all of these harmonics to the output of the synthesizer. For each of these harmonics, you can then adjust two variables: the amplitude of the harmonic, and its phase (whether it is in synch or out of synch with the original waveform).

 As Joseph Fourier showed us last time, you can create approximations of any other wave by assembling harmonics in this way.

 animation of a Fourier Series approximation of a sawtooth wave, public domain gif by Jacopo Bertolotti

You can try this at home with the updated Making Waves simulation in the PhET Collection at the University of Colorado. This simulator works much the same way as our demonstration, allowing you to select the amplitude of each harmonic, and display them both individually and in sum. Try it here:

In the top third of the screen, you set the amplitude of each harmonic. The middle third shows graphs of each harmonic, and the bottom third shows the sum of all of them. Try building a square wave, or a sawtooth wave, and see how close you can get!