Any shape from sines

A square wave has flat tops and vertical cliffs. A sine wave has neither. Yet add enough sine waves, each at the right height, and the sum draws the square — with one small spike beside each cliff that gets narrower as you add terms but never shrinks away to nothing.

What a Fourier series is

A Fourier series writes a repeating function as a sum of sines and cosines. Their frequencies are whole-number multiples of one base frequency. Each term has a weight, called its coefficient. With the right weights, the sum rebuilds the function.

Why it matters

Sines are the easy case. A spring, a circuit or a string responds to a single sine in a simple, known way. If any repeating input can be split into sines, you can work out the response to each one and add the answers. That is how Orloff's MIT differential-equations notes use the idea: to handle a system pushed by any periodic force, not just a sinusoidal one.

Building a square wave

Take a wave that sits at +1 for half its period and −1 for the other half, repeating every 2π. Flipping t to −t flips its sign, so it needs only sines. Working out the weights gives 4/(nπ) to sin nt when n is odd, and zero when n is even:

sq(t) = (4/π)(sin t + sin 3t/3 + sin 5t/5 + …)

One term is a single smooth hump: the right size, the wrong shape. Adding sin 3t/3 flattens the top and steepens the sides. Each later term is smaller and fixes finer detail. Orloff plots the sums up to n = 1, 3, 9 and 21, and the square comes into focus.

What "converges" means here

The square wave is piecewise smooth: smooth except at isolated jumps. For such a function the series converges to the function at every point where it is continuous. At a jump it converges to the midpoint of the jump. Here the midpoint is 0, and every partial sum already equals 0 at t = 0, because every sin nt is zero there. The series does not fail at the jump. It lands on the average of the two sides.

How fast the weights shrink tells you how rough the function is. A jump makes them fall like 1/n, as 4/(nπ) does. A corner, like the tip of a triangle wave, makes them fall like 1/n². A smooth function's weights fall like 1/n³ or faster. A jump needs the most help from high frequencies.

The spike that will not go away

Next to the jump, something stubborn shows up. The partial sums overshoot the flat top. Once there are more than a few terms, the peak sits at about 1.18 before the sum settles toward 1. Add more terms and the peak slides closer to the jump, but its height does not fall away toward 1. This is Gibbs' phenomenon. At any jump the overshoot is about 9% of the size of the jump. The square wave jumps from −1 to +1, a jump of 2, so the overshoot is about 0.18. Mystilidis, Tserkezis and Fikioris show it settling at 8.95% of the jump as the number of terms grows.

This does not contradict convergence. Pick any fixed point just to the right of the jump. As terms are added, the peak moves inward past it, and the sum at your point settles to 1. The overshoot never stays at one place; it rides on a peak that keeps moving toward the jump. Every point converges, and the tallest point of each finite sum stays tall. Both are true at once.

In short

A piecewise-smooth repeating shape is a sum of sines and cosines at whole-number multiples of one frequency. The smoother the shape, the faster the weights shrink. At a jump the series lands on the midpoint, and Orloff’s notes put the overshoot at “about 9%” of the jump for a sum of any number of terms; with only a handful of terms it is larger — 13.7% at one term, by our own sums — settling toward 9% as terms are added, in a spike that narrows toward the jump but never shrinks away.