Take two complex exponentials and add them.

e

^{i*a*t}+ e

^{i*b*t}

If one is much faster than the other (b>>a), you get a circle that slowly "marches around". It runs around with angular frequency "b", while its center is also going in a slow circle with angular frequency "a". This is the picture above.

If the two frequencies are equal, they would just trace out a circle of radius 2. But if they are close but slightly different, they'll drift from being in phase to being out of phase and back again. They'll almost trace out a circle of radius 2, but just barely miss because by the time they get back to where they started, they're a little bit off from each other. The result is a gradual inward spiral, shown below.

You could see this analytically like so:

e

^{i*a*t}+ e

^{i*b*t}

e

^{i*t*(a+b)/2}(e

^{i*t*(a-b)/2}+ e

^{i*t*(b-a)})

e

^{i*t*(a+b)/2}2Cos[t*(a-b)/2]

So the sum of two complex exponential runs around in a circle at the average of their frequencies, but the radius of that circle varies up and down sinusoidally in time. The projection of this complex exponential onto one of the axes describes beats - the "wa-wa-wa-wa-wa" sound you hear when two clarinet players tune.

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