---
title: "Normal Modes of a Bounded System"
description: "Fixing the ends of a string or closing a pipe allows only those wavelengths that fit, so the system rings at a fundamental and its harmonics. Which harmonics appear is set by the boundary conditions."
canonical: https://lightmysky.com/learn/science/normal-modes-of-a-bounded-system-mt_650O-YrqJs
source: https://lightmysky.com/learn/science/normal-modes-of-a-bounded-system-mt_650O-YrqJs.md
retrieved: 2026-09-12
---

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# Normal Modes of a Bounded System

Fixing the ends of a string or closing a pipe allows only those wavelengths that fit, so the system rings at a fundamental and its harmonics. Which harmonics appear is set by the boundary conditions.

Subject: Science · Area: Waves, Light & Sound · Ages 18 to 20
Page: https://lightmysky.com/learn/science/normal-modes-of-a-bounded-system-mt_650O-YrqJs

## Ready when they can

- Finds the allowed wavelengths and frequencies for a string fixed at both ends
- Compares open and closed pipes and explains the missing harmonics of a closed pipe
- Explains why the mixture of harmonics gives an instrument its timbre

## Lesson: Only the wavelengths that fit can ring

A string fixed at both ends only sings wavelengths that fit, with still nodes at each end. The rule is lambda equals 2L over n. Nodes never move, and antinodes swing widest halfway between nodes.

The nth harmonic sounds at n times the fundamental, and the fundamental is the lowest. An open pipe gives every harmonic, but a pipe closed at one end gives odd ones only: the closed end must be a node and the open end an antinode, which fits only odd quarter waves. The closed fundamental is therefore an octave below an open pipe of the same length.

**Example.** A string with a 100 hertz fundamental rings at 100, 200, and 300 hertz, each a whole multiple of the base. The bars below show those first three harmonics climbing evenly.

**Tip.** Timbre is the recipe of harmonics above the fundamental. Change the mix and the same note sounds like a new instrument.

**Recap.** Only fitting wavelengths ring, harmonics stack as whole multiples, and their mix is the timbre.

## Practice

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

## Needs first

- [Superposition and Stationary Waves](https://lightmysky.com/learn/science/superposition-and-stationary-waves-mt_AIxA2TYGJN)