The big idea: Pluck a guitar string or blow across a bottle and it only 'sings' at certain special frequencies — its harmonics.
Resonance and the first harmonic
- At those frequencies a standing wave fits neatly into the length, and the sound is loud. This is resonance.
- The lowest of these is the first harmonic.
- The others are whole-number multiples of it.
Three new words: Node = a point on a standing wave that never moves.
Antinode = a point that swings with the biggest amplitude.
Resonance = when a system is driven at one of its natural frequencies and vibrates strongly.
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What the ends force: A fixed end of a string (or the closed end of a pipe) must be a node.
A free/open end must be an antinode.
The pattern that fits between those ends decides the wavelength.
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Count how many loops (half-wavelengths) fit into the length, and you have the wavelength. Which condition you use depends on the ends.
| Boundary | Ends are… | Wavelength condition | Which harmonics |
|---|---|---|---|
| String, both ends fixed | node — node | λ = 2L ÷ n | all: n = 1, 2, 3, … |
| Pipe open at both ends | antinode — antinode | λ = 2L ÷ n | all: n = 1, 2, 3, … |
| Pipe closed at one end | node — antinode | λ = 4L ÷ n | odd only: n = 1, 3, 5, … |
| String, one end fixed, one end free | node — antinode | λ = 4L ÷ n | odd only: n = 1, 3, 5, … |
| String, both ends free | antinode — antinode | λ = 2L ÷ n | all: n = 1, 2, 3, … |
| Pipe closed at both ends | node — node | λ = 2L ÷ n | all: n = 1, 2, 3, … |
Why a closed pipe has only odd harmonics
★ Must memorise
★ Must memorise
- wavelength of that harmonic (m)
- length of the string or pipe (m)
- harmonic number (1, 2, 3 … ; for a closed pipe only odd: 1, 3, 5 …)
These two are NOT in the data booklet: You have to know λ = 2L/n and λ = 4L/n — they are not given.
Memory aid: a string or open pipe matches at its ends (both same type), so a half-wave fits → 2L/n. A closed pipe is lopsided (node one end, antinode the other), so a quarter-wave fits → 4L/n, odd harmonics only.
Then turn wavelength into frequency: Once you have the wavelength, the frequency comes from the given wave equation v = fλ (rearranged f = v ÷ λ), where v is the speed of the wave (the speed of sound for a pipe).
- wave speed (m s⁻¹)
- frequency (Hz)
- wavelength (m)
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IB-style questionCalculate[3 marks]
A guitar string of length 0.65 m is fixed at both ends. A wave travels along it at 260 m s⁻¹. Find the wavelength and frequency of its first harmonic.
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Natural frequency and resonance: Anything that can oscillate has a natural frequency f₀: the frequency it oscillates at when it is displaced and left alone.
A periodic driving force of frequency f makes it oscillate at f. The amplitude is largest when f = f₀: this is resonance, where the driver transfers energy to the system most efficiently.
Amplitude against driving frequency
What damping does: Damping is a force such as friction or air resistance that takes energy out of an oscillation.
More damping gives a lower maximum amplitude and a broader peak, and the peak moves to a slightly lower frequency than f₀.
Light, critical and heavy damping
| Damping | After the object is released | Example |
|---|---|---|
| Light | many oscillations; the amplitude falls slowly; the period hardly changes | a pendulum swinging in air |
| Critical | back to equilibrium in the shortest time, without oscillating | a car's suspension |
| Heavy | no oscillation; a slow return to equilibrium | a pendulum in thick oil |
Useful resonance
- A musical instrument: the string or air column resonates, so the note is loud.
- Tuning a radio: the circuit resonates at the frequency of the chosen station.
Destructive resonance
- A bridge sways strongly when walkers or wind drive it near its natural frequency. London's Millennium Bridge was closed two days after it opened in 2000 and later fitted with dampers.
- A building shaken by an earthquake at its natural frequency; a machine part that rattles at one engine speed.
Exam wording: At resonance the driving frequency equals the natural frequency and the amplitude is a maximum.
Engineers add damping to cut the amplitude at resonance.
How this is tested — harmonics show up as short calculations and 'read the pattern' questions:
Paper 1A
- A quick calculation — e.g. a pipe closed at one end of a given length and sound speed: find the first two harmonic frequencies.
- Pick the response curve of a heavily damped oscillator, or the true statement about light damping (the frequency does not change as it dies away).
Paper 2
- Determine a frequency by measuring the spacing of nodes/antinodes (e.g. melted spots in a microwave) and using v = fλ.
The classic trap: A closed pipe has only odd harmonics — its 'second harmonic' is actually n = 3, not n = 2.
Closed pipe — odd harmonics only: For a pipe closed at one end, λ = 4L/n with n = 1, 3, 5, …
So the first harmonic is n = 1 and the next one is n = 3 (there is no n = 2). The frequencies go in the ratio 1 : 3 : 5 …
IB-style questionCalculate[4 marks]
A pipe of length 0.20 m is closed at one end and open at the other. The speed of sound in the air inside is 340 m s⁻¹. Find the frequencies of its first two harmonics.
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