Why Radio Stations Don't Interfere

Multiply two sine waves of different whole-number frequencies and average over a period: you get exactly zero. Waves like that are orthogonal, and it's what lets a receiver pick one signal out of many sharing the air.

How it works

Periodic and orthogonal functions

In one sentence

Functions $f$ and $g$ are orthogonal on $[-L, L]$ if $\int_{-L}^{L}f(x)g(x)\,dx = 0$ ; the functions $\sin(n\pi x/L)$ , $n \ge 1$ , are mutually orthogonal there, with $\int_{-L}^{L}\sin^2(n\pi x/L)\,dx = L$ .

General formula

\int_{-\pi}^{\pi}{\htmlData{sym=2}{\sin(mx)}}\,\sin({\htmlData{sym=1}{n}}x)\,dx = \begin{cases}0 & {\htmlData{sym=0}{m}} \ne {\htmlData{sym=1}{n}}\\ \pi & {\htmlData{sym=0}{m}} = {\htmlData{sym=1}{n}}\end{cases}

For $m \ne n$ the two waves drift in and out of step equally, so every stretch where their product is positive is matched by one where it's negative.

What the symbols mean

$m$
First frequency
How many cycles the first wave makes over the period.
$n$
Second frequency
How many cycles the second wave makes.
$\sin(mx)$
Carrier m
One station's wave.

Two sine waves of different whole-number frequencies, multiplied and integrated over a period, cancel exactly: positive and negative areas match.
Only when $m = n$ does the product stay non-negative, giving area $\pi$ .
So $\sin x, \sin 2x, \sin 3x, \dots$ are orthogonal, like perpendicular axes for functions.
A receiver multiplies the incoming signal by its own carrier and integrates: every other carrier integrates to zero.

Learn the math

Periodic and Orthogonal FunctionsDiff Eq · Lesson 43, Boundary Value Problems and Fourier Series