---
title: "Fourier Analysis and the Wave Packet"
description: "Any repeating shape is a sum of sines, and a pulse is a sum over a continuous range of frequencies. The shorter the pulse, the wider the range of frequencies it needs."
canonical: https://lightmysky.com/learn/science/fourier-analysis-and-the-wave-packet-mt_1ZaojWUZEw
source: https://lightmysky.com/learn/science/fourier-analysis-and-the-wave-packet-mt_1ZaojWUZEw.md
retrieved: 2026-09-12
---

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# Fourier Analysis and the Wave Packet

Any repeating shape is a sum of sines, and a pulse is a sum over a continuous range of frequencies. The shorter the pulse, the wider the range of frequencies it needs.

Subject: Science · Area: Waves, Light & Sound · Ages 21 to 22
Page: https://lightmysky.com/learn/science/fourier-analysis-and-the-wave-packet-mt_1ZaojWUZEw

## Ready when they can

- Writes a square wave as a sum of harmonics and describes what adding more terms does
- Relates the width of a pulse in time to the spread of frequencies it contains
- Applies the idea to bandwidth in a communication channel

## Lesson: Waves built from waves

Any repeating shape is a sum of sines at whole multiples of a base frequency. A square wave, for instance, is built from odd multiples stacked together. Superposition does the work: waves add point by point wherever they meet.

Each new harmonic sharpens the corners of the square wave sum. In the limit the sum reproduces the jumps of the square wave. More terms always mean a closer match.

**Example.** Squeeze a pulse shorter in time and it demands a wider range of frequencies. A needle thin spike needs a broad band, while a long gentle swell needs only a narrow one. Short in time means wide in frequency.

**Tip.** A channel must carry every frequency its signal contains. A narrow bandwidth rounds off fast changes, so sharp pulses arrive smeared. Match the bandwidth to the fastest change you must preserve.

**Recap.** Sums of sines build any shape, and shorter pulses always cost a wider band.

## Practice

8 questions on this page, each with its working shown.

## Needs first

- [Fourier Series](https://lightmysky.com/learn/mathematics/fourier-series-mt_DJ3iwfj7NK)
- [The Uncertainty Principle from Wave Packets](https://lightmysky.com/learn/science/the-uncertainty-principle-from-wave-packets-mt_DPYnoGksRd)
- [X-Ray Diffraction and the Bragg Condition](https://lightmysky.com/learn/science/x-ray-diffraction-and-the-bragg-condition-mt_qEDplGeIkB)

## Opens up

- [Stimulated Emission and How a Laser Works](https://lightmysky.com/learn/science/stimulated-emission-and-how-a-laser-works-mt_xgjD0mzEAV)
- [Time-Dependent Perturbation Theory and Fermi's Golden Rule](https://lightmysky.com/learn/science/time-dependent-perturbation-theory-and-fermis-golden-rule-mt_zDSASGF7SC)