Adding sines.

Step 1 of 6

One circle

In the trigonometry Unit you saw a point turning on a circle draw a sine wave. In complex numbers: $b e^{ikt}$⁠ is a point turning $k$⁠ times as fast, on a circle of radius $b$⁠, and its height is $b\sin kt$⁠. Add several of these, each circle on the rim of the last, and the tip of the chain draws a sum of sines. Joseph Fourier claimed that every repeating wave can be built this way:

$f(t) = b_1 \sin t + b_2 \sin 2t + b_3 \sin 3t + \dots$$f(t) =$$b_1 \sin t + b_2 \sin 2t + b_3 \sin 3t + \dots$⁠

This is a Fourier series. Here the wave is a square wave, and there is only one circle: the first term of its series, $\frac{4}{\pi}\sin t$⁠. Work out the circle’s radius.

Step 2 of 6

More circles

The series of the square wave, $s$⁠, has only odd frequencies, and each term shrinks like one over its frequency:

$s(t) = \frac{4}{\pi}\left(\sin t + \frac{\sin 3t}{3} + \frac{\sin 5t}{5} + \dots\right)$$s(t) =$$\frac{4}{\pi}\left(\sin t + \frac{\sin 3t}{3} + \frac{\sin 5t}{5} + \dots\right)$⁠

Here there are $4$⁠ circles, and Show the sines draws each sine on its own on the graph. Predict what the fifth circle will add.

Step 3 of 6

The triangle wave

With Wave on the triangle, $g$⁠, the terms shrink faster, like one over the square of the frequency:

$g(t) = \frac{8}{\pi^2}\left(\sin t - \frac{\sin 3t}{9} + \frac{\sin 5t}{25} - \dots\right)$$g(t) =$$\frac{8}{\pi^2}\left(\sin t - \frac{\sin 3t}{9} + \frac{\sin 5t}{25} - \dots\right)$⁠

A smoother wave needs fewer fast sines. Work out the radius of the third circle.

Step 4 of 6

Beside the jump

The square wave jumps all at once from $-1$⁠ to $1$⁠. A sum of smooth sines cannot jump like that, so beside every jump it rises a little above the wave, then wobbles. Show the sines is now at the end, and the board shows the bump’s height. Predict what will happen to it as you add Terms.

Step 5 of 6

No jump

The bump belongs to jumps, not to the wave itself. Try all three choices of Wave, then answer.

Step 6 of 6

The highest point

Under Show the sines you see the sum up close beside the jump.