AS 91523 · 2024 paper · the pan flute

Question 1(c) Merit to Excellence

A pan flute is a set of pipes of different lengths bound together, each CLOSED at one end and open at the other. Sound is made by blowing across the open ends. Jane's shortest pipe is 8.50 cm long and her longest is 17.5 cm. The speed of sound in air is 343 m s⁻¹.

v sound	=	343 m s⁻¹
shortest pipe	=	8.50 cm
longest pipe	=	17.5 cm

Explain how standing waves form in a pan flute pipe, open at one end and closed at the other, and explain why not all harmonics can form.

The 2024 paper, page 2 ↗ Marking schedule, page 1 ↗

See it move: Standing waves: string, open pipe, closed pipe

Work it out

Try it on paper first. Stuck? Take one hint at a time.

1 Where do the waves that build the standing wave come from?
Blowing across the open end sends a sound wave down the pipe. It reflects at the closed end, and again at the open end, so waves of the same frequency and amplitude are travelling up and down the pipe at once.
2 What happens where those two waves overlap?
They superpose. At some points they are always 180° out of phase and cancel: nodes, where the air never moves. Halfway between, they are always in phase and add: antinodes, where the air moves most. The pattern stays put, so it is a standing wave.
3 What has to happen at the closed end, and at the open end?
At the closed end the wall stops the air moving, so that end must be a node (the reflection there turns the wave over, 180°). At the open end the air is free to move, so it must be an antinode (no phase change on reflection).
4 Which patterns fit, with a node at one end, an antinode at the other?
A node to the next antinode is a quarter-wavelength, so the pipe must hold an odd number of quarters: L = λ/4, 3λ/4, 5λ/4, and so on. Those are f₁, 3f₁, 5f₁: the odd harmonics.
5 So why is there no second harmonic?
Two quarter-wavelengths make a half-wavelength, which runs from a node to a node or from an antinode to an antinode: the same kind of point at both ends. This pipe’s ends are different, so the 2nd, 4th and every even harmonic cannot form.

Answer Sound reflects off both ends of the pipe, and the waves going each way superpose: always out of phase at the nodes (no movement), always in phase at the antinodes (most movement). The closed end must be a node, since the air cannot move against the wall, and the open end an antinode, since the air moves freely there. Only an odd number of quarter-wavelengths fits between a node and an antinode (L = λ/4, 3λ/4, 5λ/4, …), so only the odd harmonics form; an even one would need the same kind of point at both ends.

What the marker wanted

Achieved One of: the waves reflect off the ends and superpose; in-phase waves give an antinode; waves 180° out of phase give a node; or that a closed pipe needs a node where it is closed and an antinode where it is open.
Merit Gives the reasons for the end conditions, or explains why only an odd number of quarter-wavelengths gives a node where the pipe is closed and an antinode where it is open.
Excellence Both: the reasons for the end conditions AND the odd quarter-wavelengths. An explanation in terms of pressure is accepted too, if it is right.

In our words, from NZQA’s 2024 marking schedule.

Common mistake

Explaining the nodes and antinodes with a path difference, as if there were two sources. NZQA’s 2024 report lists this among answers that did not reach Achieved: a standing wave is one wave and its own reflection, travelling opposite ways.

How did it go?

Got it moves it further out. Review brings it back tomorrow.