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Illustration 16.5: The Fourier Series, Qualitative Features

n =
n =

Please wait for the animation to completely load.

We have thus far only looked at simple periodic motion that can be described by a single sine or cosine. This may seem like a horrible mistake. Most periodic functions are tremendously complicated. Have we been doing something wrong by focusing on only sines and cosines? Well, actually not. ANY periodic function can be represented as a sum of sines or cosines!   We find that sometimes we may need an infinite number of them, but nonetheless we are able to describe any periodic phenomena, no matter how complicated, in this way. Restart.

Consider a sawtooth (position is given in meters) function that is periodic with L = 1 (it is shown over two periods since it is easier to see the function this way). In the animation the amplitude is a function of x, but it could have been a function of time. Select "play the Fourier series of the sawtooth." The gray function is the actual sawtooth, while the red function is the total approximate sawtooth from a Fourier series (if you did not change n, the animation shows the n = 1 term only). Change n, the number of sine functions that will be added together to approximate the sawtooth, and see how the red function changes. The green sine function is the current term that is added to get the total red function. On the right is a representation of the relative amount of each sine function as it is added to the total. You may add as many as 35 terms. Also note that at the point where the sawtooth kinks, there is always overshoot (this is called the Gibbs phenomenon).

Now look at the square wave. It turns out that the n = 2, 4, 6, ... terms do not contribute to the sum. Verify this by watching the animation for n = 35.

When you get a good-looking graph, right-click on it to clone the graph and resize it for a better view.

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