TwiddleLab / Physics / Waves and Sound

Standing Waves: Why Strings Have Notes

Lesson 7 of 10 · 7 min · Free, no account needed

What you will learn

• A string fixed at both ends can only hold standing waves with a whole number of loops: λ = 2L / n.
• Nodes never move; the loops between them swing. The patterns are the harmonics f, 2f, 3f...
• Pressing a fret shortens the vibrating string, so the note goes up: half the length, twice the frequency.

Start the lesson

How the lesson goes

1. A guitar string is one string. Why does pressing a fret change the note?

   First a guess: where would you press to play the same note an octave higher?

2. Shake a stretched cord from one end. Find the frequencies where it stands still in loops.

3. Now a guitar string. Find the note an octave higher than the open string.

   The lowest pattern on a string is one loop, so its wavelength is twice the vibrating length. Halve the length and the wavelength halves; the wave speed on the string stays the same, so f = v/λ doubles.

4. Standing waves.

   A standing wave forms when a wave and its reflection add up on a string fixed at both ends. The loops swing; the pattern does not travel.

   Nodes never move. Between them are loops. A string of length L fits n loops when λ = 2L / n.

   These patterns are the harmonics: f, 2f, 3f... A plucked string plays them all at once, which gives each instrument its sound.

   Simplified: an ideal, perfectly flexible string. Real strings are a little stiff, so high harmonics come out slightly sharp.

5. Tap every node of this pattern.

   A node is where the wave and its reflection always cancel. With 3 loops the string is cut into thirds, so nodes sit at 0, L/3, 2L/3 and L.

6. Game: Play the riff. Hit each note by placing your finger.

   Your score is the notes you hit on the first try.

7. AI: a computer hears sound as fingerprints of frequencies. Which ones are strings?

   A plucked string vibrates in many standing-wave patterns at once, at f, 2f, 3f... A bell's patterns are not whole-number multiples, and a whistle is almost one pure tone.

8. Place your finger to play a note 1.5 times higher than the open string.

   The wave speed on the string does not change when you press it, and the vibrating part holds one loop, so f = v / (2 × length). Make the length 2/3 and f becomes 3/2.

9. Your riff, played back.

Previous lesson: Waves Add Up: Interference · Next lesson: Resonance: Pushing at the Right Time · Part of the course: Waves and Sound

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