Fourier Sine Series

A Fourier sine series is simply a way to build almost any shape on an interval by stacking up sine waves, the way a musical chord is built from notes. (It's named after Joseph Fourier, who used these sums to work out how heat spreads through solid objects.) It's how you'll split the starting shape of a plucked string or a heated rod into pieces that each behave simply. The formula $f(x) = \sum B_n\sin\frac{n\pi x}{L}$ looks heavy, but it's just a recipe: $\sin\frac{n\pi x}{L}$ (the wave with $n$ half-bumps between $0$ and $L$) and $B_n$ (how much of that wave goes in). The formula for $B_n$ is the "multiply and integrate" trick from the last lesson, which picks out one wave at a time. It's a few simple parts put together, so bear with it.

The eigenfunctions $\sin\frac{n\pi x}{L}$ are the shapes a string fixed at both ends can take on its own. So here's the natural question: can any starting shape $f(x)$ on $[0, L]$ be built out of them? It can, and orthogonality tells you how much of each one to use.

Multiply both sides by $\sin\frac{m\pi x}{L}$ and integrate over $[0, L]$: orthogonality wipes out every term but $n = m$, leaving the formula for $B_m$. That one surviving term is $B_m\int_0^L\sin^2\frac{m\pi x}{L}\,dx = B_m\cdot\frac L2$, half of the $L$ you'd get over $[-L, L]$. Dividing by $\frac L2$ is where the $\frac2L$ in front comes from.

Why does the series describe an odd function? Every sine is odd and repeats every $2L$, so any sum of them is too. On $[0, L]$ the series matches $f$, and on $[-L, 0]$ it automatically gives $-f(-x)$, the odd extension.

Example: $f(x) = x$ on $[0, L]$. Integration by parts gives $B_n = \frac{2L(-1)^{n+1}}{n\pi}$, so

A common mistake. It's tempting to think this series equals $L$ at $x = L$, since $f(L) = L$. It isn't: every $\sin\frac{n\pi x}{L}$ is $0$ at $x = L$, so the whole series is $0$ there. The odd extension jumps from $L$ down to $-L$ at that point, and the series lands in the middle.

The coefficients shrink like $\frac1n$ when $f$ has a jump (in its odd periodic extension); smoother functions have faster-shrinking coefficients. It makes sense: a sharp jump needs lots of high-frequency sines to build, so they can't fade away quickly. The graph further down shows the partial sums for $f(x) = x$ closing in everywhere except at that jump.

Sine series solve the heat equation in a rod with ends held at zero and the wave equation for a plucked string: each sine term evolves on its own.