Waves · Quick Reference

Standing Waves Cheat Sheet

Waves · Lesson 3/7

In one line: a standing wave forms when two identical waves traveling in opposite directions interfere, creating a pattern that appears to oscillate in place rather than travel — the physics behind every musical instrument.

Key Ideas

1How Standing Waves Form. Created when a wave and its reflection (traveling in opposite directions) interfere, producing a stable pattern of nodes and antinodes rather than a traveling wave.

2Nodes and Antinodes. Nodes are points that stay completely still (zero displacement); antinodes are points that oscillate with maximum amplitude, located exactly between adjacent nodes.

3Standing Waves on a String (Fixed at Both Ends). Only specific wavelengths 'fit' on a string fixed at both ends: lambda_n = 2L/n, where L is the string length and n is a positive integer (the harmonic number).

4Fundamental Frequency and Harmonics. The fundamental (first harmonic, n=1) is the lowest possible frequency; higher harmonics (n=2,3,4...) are whole-number multiples of the fundamental frequency.

5Standing Waves in Musical Instruments. String instruments (guitars, violins) and wind instruments (flutes, organs) both rely on standing wave patterns to produce their characteristic musical pitches.

Worked Examples

A string of length 2m is fixed at both ends. Find the wavelength of its fundamental (n=1) standing wave.

4 meters

A string's fundamental frequency is 100 Hz. Find the frequency of its 3rd harmonic.

300 Hz

What is located exactly halfway between two adjacent nodes on a standing wave?

An antinode

Formulas

lambda_n = 2L/n (string fixed at both ends)

f_n = n*f_1 (harmonics are multiples of the fundamental)

Practice Yourself

A string of length 3m is fixed at both ends. Find the fundamental wavelength.

6m (using 2L/1)

A string's fundamental frequency is 50Hz. Find its 2nd harmonic frequency.

100Hz

What is a 'node' in a standing wave?

A point of zero displacement (stays still)