Harmonics, resonance and wavelength from a standing-wave pattern
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Question
What is a node?
Answer
A point on a standing wave that **never moves** (zero amplitude).
Question
What is an antinode?
Answer
A point on a standing wave that swings with the **largest** amplitude.
Question
What is the first harmonic?
Answer
The **lowest** frequency at which a string or air column resonates — the standing-wave pattern with the fewest loops.
Question
What is resonance?
Answer
When a system is driven at one of its **natural frequencies** and vibrates with a large amplitude — that is what makes a harmonic loud.
Question
Wavelength condition for a string fixed at both ends (or a pipe open at both ends)?
Answer
**λ = 2L/n** for n = 1, 2, 3, … — n half-wavelengths fit into the length L.
Question
Wavelength condition for a pipe closed at one end?
Answer
**λ = 4L/n** with **n = 1, 3, 5, …** (odd harmonics only — node at the closed end, antinode at the open end).
Question
Are λ = 2L/n and λ = 4L/n in the data booklet?
Answer
**No** — you must memorise them. Only the wave equation v = fλ is given.
Question
Why does a pipe closed at one end have only odd harmonics?
Answer
Its ends are different (node at the closed end, antinode at the open end), so only odd numbers of quarter-wavelengths fit: λ = 4L/n, n = 1, 3, 5, …
Question
How far apart are two neighbouring nodes (or antinodes)?
Answer
**Half a wavelength.** So λ = 2 × the node-to-node spacing.
Question
How do you turn a wavelength into a frequency?
Answer
Use the given wave equation **v = fλ**, rearranged to **f = v ÷ λ** (v is the wave speed — the speed of sound for a pipe).
Question
A 0.65 m string fixed both ends, wave speed 260 m s⁻¹ — first-harmonic frequency?
Answer
λ = 2L = 1.3 m; f = v/λ = 260/1.3 = 200 Hz.
Question
How can melted spots in a microwave give the microwave frequency?
Answer
The spots (antinodes) are half a wavelength apart; double the spacing for λ, then f = c/λ.
Question
What does increasing the damping do to a driven oscillator's resonance peak?
Answer
The maximum amplitude **falls**, the peak gets **broader**, and it moves to a **slightly lower** frequency.
Question
Light, critical and heavy damping: what happens after release?
Answer
Light: many oscillations, the amplitude dies slowly. Critical: back to equilibrium in the **shortest time** without oscillating. Heavy: no oscillation, a slow return.
Question
Give a useful and a destructive effect of resonance.
Answer
Useful: a musical instrument sounds loud; tuning a radio. Destructive: a bridge or building shaken strongly at its natural frequency.
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Topic 3.4 hub
Standing waves and resonance
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