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📘 Lesson 3 of 7 · Waves

🎸 Standing Waves

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.

Course progress: 43%

01 Key Concepts

How 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.

Nodes and Antinodes

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

Standing 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).

Fundamental 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.

Standing 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.

A standing wave pattern with nodes and antinodes

02 Key Formulas

03 Solved Examples

Example 1 A string of length 2m is fixed at both ends. Find the wavelength of its fundamental (n=1) standing wave.
  1. Apply lambda_n = 2L/n with n=1: lambda_1 = 2*2/1.
Answer: 4 meters
Example 2 A string's fundamental frequency is 100 Hz. Find the frequency of its 3rd harmonic.
  1. Apply f_n = n*f_1 with n=3: f_3 = 3*100.
Answer: 300 Hz
Example 3 What is located exactly halfway between two adjacent nodes on a standing wave?
  1. Between adjacent nodes (still points), the wave reaches its maximum oscillation.
  2. This point of maximum oscillation is a specific named feature.
Answer: An antinode

04 Practice Questions

1A string of length 3m is fixed at both ends. Find the fundamental wavelength.
6m (using 2L/1)
2A string's fundamental frequency is 50Hz. Find its 2nd harmonic frequency.
100Hz
3What is a 'node' in a standing wave?
A point of zero displacement (stays still)
4What is an 'antinode'?
A point of maximum displacement (oscillates most)
5What two things combine to create a standing wave?
A wave and its reflection traveling in opposite directions

📄 Standing Waves — Downloadable Worksheet

10 questions with a full answer key. Grab the PDF to print, or try the interactive version in your browser.

🗒 Cheat Sheet ⬇ Download PDF 🖥 Interactive Worksheet