Saturday, November 7, 2009

Fourier analysis 1 - Sound waves

First, I need to apologize for the lack of activity on this blog, and regretfully state that the relative dearth of new posts will likely continue for another few months. I'm at the point of my career when I try to graduate and get a job for next year, and between these two activities I don't have much time for posting to this blog. I do plan on continuing it, but it will of necessity be sporadic for a few more months.

Now that that is out of the way, I want to discuss Fourier analysis, which I mentioned at the end of the last post (over two months ago). One theme that may have come through to someone reading this blog since the beginning is the ubiquity of "waves" in physics. When discussing the Doppler effect back in March, I used sound as an example (the police siren) before moving to light. I want to do the same thing now. Sound is a pressure wave that moves through the air and is interpreted by our ears. Just as the color of light is determined by its frequency, the pitch of sound is also determined by the frequency of the sound wave. People who play music will be very familiar with this - the root A note, for example, is a sound wave with a frequency of 440 Hz (if I haven't used this unit before, a Hz is just inverse seconds. So 440 Hz means that the wave oscillates 440 times per second). Let's use the power of modern computers to show a video of me playing the A on my guitar:

The idea here is fairly simple. The guitar is tuned so that plucking the string makes it oscillate at 440 Hz, creating the note that we hear.

On the opposite end of the spectrum from a perfectly pitched musical note is "white noise." We all know what white noise is, it's static, something with no discernable pattern. It's called white because the color white is a combination of all colors. White noise is a combination of all frequencies. For a lovely example of white noise, one can go to http://simplynoise.com/.

The point of this is that waves are very well understood mathematically. Therefore, we can very easily represent these sounds with a mathematical expression. For example, the A note I played in the video can be represented as an oscillating wave with frequency 440 Hz, and it would look something like the drawing to the right. There's clearly a pattern in there of the appropriate frequency (I also added an overall envelope to describe the starting and stopping of the pulse, but that's not really important for this discussion).

White noise looks like the next plot, and there is no pattern there.

In part 2, I'll talk about what happens when you add more tones to form a chord (or an orchestra) or what happens when you add noise to a tone.