Harmonics and overtones explain why musical instruments can play rich, recognizable sounds instead of just one plain tone. When a string, air column, or other object vibrates, waves can reflect and combine to form standing waves. Only certain patterns fit the boundaries of the system, so only certain frequencies are strongly produced.
These allowed frequencies give instruments their pitch and tone color.
Understanding Physics: Harmonics and Overtones
A standing wave has places that barely move, called nodes, and places that move the most, called antinodes. These positions are set by the boundaries. A guitar string cannot move at the bridge or nut, so those points stay as nodes.
In a wind instrument, the air behaves differently at an open end and a closed end. An open end allows air motion, while a closed end blocks it. This difference explains why pipes with different end conditions produce different sets of notes.
The vibrating object does not choose every possible frequency. Its shape and boundaries select the patterns that can keep reinforcing themselves.
The lowest pattern usually carries much of the sound energy, so listeners identify it as the pitch of the note. Higher patterns vibrate faster at the same time. Their frequencies are related to the fundamental by whole-number multiples in many ideal systems.
A plucked string therefore moves in a complicated shape, not just one simple arc. It can contain a broad pattern across its full length plus shorter patterns with extra nodes.
The ear combines these frequency components into one musical sound. A tuning app may show the strongest low frequency while a spectrum display reveals several higher peaks above it.
The mix of higher frequencies gives an instrument its timbre, or sound quality. A violin and a flute can play the same named note at the same fundamental frequency, yet they sound clearly different. Their bodies, materials, and playing methods favour different harmonics.
Plucking near the middle of a string tends to suppress some patterns because that location is a node for them. Plucking closer to an end excites more high-frequency patterns and often gives a brighter sound.
Bowing, blowing, striking, and damping all change which vibrations survive. This is why a musician can change tone without changing the intended pitch.
Real instruments are not perfectly ideal. A thick stiff string has slightly shifted upper frequencies because bending stiffness affects its motion. Piano tuners allow for this effect when tuning high notes.
In wind instruments, the effective length of the air column can differ from the visible tube length because air motion extends a little beyond an opening. Tone holes, bell shapes, and mouthpieces further alter the resonances. When studying this topic, draw the nodes and antinodes before trying to predict frequencies.
Count the sections of the pattern that fit into the vibrating length. Keep the words harmonic and overtone separate.
Harmonic numbers count from the fundamental, while overtone numbers count above it. That naming difference matters most when some harmonic numbers are missing.
Key Facts
- For a string fixed at both ends, the allowed wavelengths are λn = 2L/n, where n = 1, 2, 3, ...
- For a string fixed at both ends, the harmonic frequencies are fn = nv/(2L), where v is wave speed and L is string length.
- For an open pipe, both ends are displacement antinodes, so fn = nv/(2L), where n = 1, 2, 3, ...
- For a closed pipe, one end is a displacement node and the other is an antinode, so fn = nv/(4L) for n = 1, 3, 5, ... only.
- The fundamental frequency is the first harmonic, f1, and it is the lowest allowed standing-wave frequency.
- The first overtone is not always the second harmonic. In a closed pipe, the first overtone is the third harmonic.
Vocabulary
- Harmonic
- A harmonic is an allowed standing-wave frequency that is an integer multiple of the fundamental frequency in systems such as strings and open pipes.
- Overtone
- An overtone is any allowed frequency above the fundamental frequency.
- Fundamental frequency
- The fundamental frequency is the lowest frequency at which a system can form a standing wave.
- Node
- A node is a point in a standing wave that stays still because destructive interference always occurs there.
- Antinode
- An antinode is a point in a standing wave that has maximum vibration amplitude.
Common Mistakes to Avoid
- Calling every overtone the same number as its harmonic is wrong because the first overtone is the frequency just above the fundamental, while the second harmonic is only the first overtone in systems that include all harmonics.
- Using fn = nv/(2L) for a closed pipe is wrong because a closed pipe has a node at the closed end and an antinode at the open end, allowing only odd harmonics.
- Drawing displacement nodes at the open ends of an air pipe is wrong because open ends are displacement antinodes and pressure nodes.
- Forgetting that wave speed depends on the medium is wrong because changing string tension, linear density, air temperature, or material changes the harmonic frequencies.
Practice Questions
- 1 A guitar string is 0.65 m long and wave speed on the string is 260 m/s. Find the fundamental frequency and the third harmonic frequency.
- 2 An open organ pipe is 1.20 m long. Using the speed of sound as 343 m/s, find the first three harmonic frequencies.
- 3 A closed pipe and an open pipe have the same length and contain air at the same temperature. Explain why the closed pipe has a lower fundamental frequency and why it does not produce even harmonics.