Vibrations

# 4 Fourier’s big idea

### Simple harmonic motion

Simple harmonic motion- like a weight bobbing up and down on the end of an ideal spring- is the simplest kind of vibration imaginable. What makes simple harmonic motion “simple” is the sine curve that represents the motion. SHM can be described by a single sine curve with exactly one specific frequency. Complex vibrations cannot be represented by a single sine curve.

### Many frequencies, all at once

In 1801, Fourier recognized that any periodic vibration- not matter how complex- can be made by adding sine curves together. Fourier’s basic idea is this:

Any periodic vibration- no matter how complicated- is the result of multiple simple harmonic oscillations (each with its own frequency and amplitude) happening at the same time.

It’s worth repeating: Complicated periodic vibrations are made up of many simple oscillations happening simultaneously.

Fourier’s idea means that any complex period vibration (like the black one shown below) is results from many different simple oscillations (shown in brown below) happening together and adding up:

Most musicians know this instinctively. When you play a note on an instrument, you don’t hear just a single pure tone. Listen carefully and you hear that the single musical note is actually made up of many different tones, all going on at the same time.

### Why Fourier’s idea is important

Fourier’s idea is extremely powerful when working with sound- particularly musical sound. Fourier’s idea can be used in two different ways. First, you can break down any periodic vibration into its various frequency components. This is called Fourier analysis. With Fourier analysis, you can describe any complex vibration in terms of which frequencies are present and how much of each of those frequency components is present. The math for Fourier analysis is way beyond the scope of this book, but most audio software has automatic routines for doing Fourier analysis built right in. These powerful tools provide a quick list of a sound’s “ingredients”- a list of which frequencies are present and in what amounts. Filters, mixers and other studio hardware often have independent controls for different frequency bands. By knowing a sound’s “ingredient list,” you can choose how to manipulate the sound. You can use Fourier’s idea in reverse to build sounds,too. Any complex vibration can be build by adding together vibrations with different frequencies. This process is called Fourier synthesis.

### Fundamental and overtones

The lowest frequency present in a vibration is called the fundamental frequency or the fundamental (for short). This is the frequency that corresponds to the frequency of repetition of the overall pattern, no matter how simple (or complex) the pattern is. Most vibrations also contain other, higher frequencies, called overtones, in addition to the fundamental frequency.

If the overtones are at integer multiples of the fundamental frequency, the overtones are usually called harmonics. For instance, if a sound contains frequencies of 100 Hz, 200 Hz, 300 Hz and so on, the fundamental is 100 Hz and the other frequencies are called harmonics. If the sound contains frequencies of 100 Hz, 2356 Hz and 4703 Hz, the fundamental is 100 Hz, the two overtones (2356 Hz and 4703 Hz) are not called harmonics.

### Fundamental frequency from a time domain graph

For complex vibrations, the fundamental frequency can be found by finding the fundamental period from the time graph. The time for one full cycle  is called the fundamental period. The word “fundamental” refers to the fact that this is the period for the overall pattern. You can find fundamental period (or frequency) from a time domain graph in pretty much the same way as you find period (or frequency) for a simple harmonic oscillator. For period, measure the horizontal “distance” for one full cycle of the vibration. For frequency, divide the number of cycles by the amount of time it takes to complete those cycles.

The fundamental period is the longest of the many different periods of vibration present in a complex vibration. The fundamental period is also the most important for sound- this is the period that the human brain uses to build the sensation of pitch.

### Online resources

Play with this HTML5 Fourier synthesizer Use the horizontal slider at the bottom of the screen to adjust the fundamental frequency. Use the vertical sliders to adjust the amounts of various overtones. Watch and listen as the graph and the sound change to match your choices.

Watch Coupled Oscillator  (0:26 youTube video) to see how it’s possible for a system to have two frequencies at the same time. 