1564. I'm a musician seeking to better understand harmonics. I've read plenty of definitions and so I understand "what" they are, but I'm having a hard time visualizing them. What confuses me is the idea that a string can vibrate at multiple frequencies at once, that it can support multiple standing waves. How could a string bend in so many different ways at the same time? thanks!! — K, Mountain View, California
To help you visualize how a string can vibrate at several frequencies at once, I wrote a flash program that shows you what a vibrating string looks like. That program should appear below this note. It allows you to adjust eight parameters: the amplitudes of the string's four simplest vibrational modes (its fundamental vibration through its fourth harmonic vibration) and the phases of those modes. The program starts with a pure fundamental vibration of the string, which is easy to visualize. But you can turn on the second, third, and fourth harmonic vibrations to whatever extent you like. What you'll observe is that a string that's vibrating at several frequencies at once has a complicated shape, but doesn't look all that unfamiliar. It's simply a mixture of several standing waves that evolve at different rates. As a result, it exhibits a fancy rippling shape that you've probably see on a jump rope or a clothesline.
If you look carefully at the string while it's vibrating in a mixture of several harmonics, you'll see that it has only one shape at any moment in time. It's just a jiggling string, after all. The parts of that shape, however, are evolving at different rates in time and those parts are actually the different harmonics going through their individual motions at their own frequencies.