The Fourier Series Toolkit

A Fourier series is simply a way of writing a shape as a sum of simple waves, sines and cosines. It's named after Joseph Fourier, who came up with it in the early 1800s while working out how heat flows. It's what will let us match any starting temperature or string shape in the units ahead, and it's the same idea behind MP3s and audio equalizers. The formula $f(x) = \sum B_n\sin\frac{n\pi x}{L}$ looks heavy, but it's just a recipe: $\sin\frac{n\pi x}{L}$ (a wave with $n$ humps across the interval), $B_n$ (how much of that wave goes in), and $\sum$ (add them all up). The coefficient formula is just a way of measuring how much of each wave $f$ contains. Once you see it as a recipe, it's a lot less scary.

Why we need this. In the next unit, separating variables in the heat equation will hand us one simple solution for each $n$, each shaped like $\sin\frac{n\pi x}{L}$ and fading in time. Superposition lets us add them up, but at $t = 0$ the sum has to match whatever starting temperature $f(x)$ we were given. So the real question is: can we write $f$ as a sum of sines, and how much of each one do we need? That's a Fourier series.

Why the coefficient formulas work. Multiply the series by $\sin\frac{m\pi x}{L}$ and integrate. Orthogonality kills every term except $n = m$, which leaves $B_m\frac L2$. Divide by $\frac L2$ and you have the formula for $B_m$. If that feels familiar, it should: it's the same move as finding a vector's component along one axis with a dot product. The perpendicular axes contribute nothing, so only the piece you want survives.

Which series? The boundary conditions decide. Every $\sin\frac{n\pi x}{L}$ is zero at $0$ and $L$, so a sine series automatically respects $u = 0$ at the ends. Every cosine has zero slope there, so cosines fit insulated ends.

Convergence. For piecewise smooth $f$, the series converges to $f$ where $f$ is continuous and to the average of the two one-sided limits at a jump. Near a jump, partial sums overshoot by about 9% no matter how many terms you take (the Gibbs phenomenon). There's a slider further down that lets you add terms and watch the overshoot refuse to go away.

Smoothness and decay. Coefficients of a function with jumps shrink like $\frac1n$; a continuous function with corners, like $\frac{1}{n^2}$. Smoother data need fewer terms. That makes sense: a sharp feature can only be built out of fast wiggles, so the high-frequency modes have to carry more weight.

Fourier series are how MP3 and JPEG compress data, how audio equalizers work, and how every separation-of-variables solution in this course is assembled.