AS 91523 · 2024 paper · the pan flute

Question 1(d) 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

Some pan flutes are tuned by moving a small cork inside a pipe to change its length. At first the longest pipe is too long, and this untuned pipe makes a beat of 9.00 Hz with a tuned pipe. The tuned pipe is 0.164 m long and plays 523 Hz on its first harmonic.

Calculate how much further the cork needs to be pushed into the pipe, and explain how you can hear when the cork is in the right position.

The 2024 paper, page 3 ↗ Marking schedule, page 2 ↗

Work it out

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

1 The untuned pipe is too long. Is its note above or below 523 Hz?
Below. A longer pipe holds a longer wavelength, and with v fixed, v = fλ gives a lower frequency.
2 What does the 9.00 Hz beat tell you about its frequency?
f_beat = |f₁ − f₂|, so the untuned pipe plays 523 − 9.00 = 514 Hz.
3 How long is the untuned pipe’s air column?
A closed pipe’s first harmonic has L = λ/4. λ = v / f = 343 / 514 = 0.6673 m, so L = 0.6673 / 4 = 0.16683 m. That is not the pipe’s full 17.5 cm, which would play 343 / (4 × 0.175) = 490 Hz: the cork already sits partway in, leaving 0.1668 m of air.
4 How far does the cork have to move?
From 0.16683 m down to the tuned 0.164 m: ΔL = 0.16683 − 0.164 = 0.00283 m, so about 2.83 mm further in. (Working from the 0.164 m and 523 Hz alone, with f × L constant, gives 2.87 mm: the 0.164 m is itself rounded.)
5 How would you hear that the cork is in the right place?
Play both pipes together while pushing the cork in. The untuned pipe’s frequency rises towards 523 Hz, so the beat frequency falls: the loud pulses come further apart. When the beats disappear, the two frequencies are equal and the pipe is in tune.

Answer About 2.83 mm (0.00283 m) further in. The untuned pipe plays 514 Hz, so its air column is 343 / (4 × 514) = 0.1668 m, against the tuned pipe’s 0.164 m. As the cork goes in the beats slow down, and when they stop the pipe is in tune.

What the marker wanted

Achieved The untuned pipe’s frequency, 514 Hz, or that no beat is heard once it is in tune.
Merit The right length for the untuned pipe, or that the beat frequency falls (longer gaps between loud pulses) as the two frequencies get closer.
Excellence The right ΔL, AND the beats slowing as the frequencies get closer, until none is heard.

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

Common mistake

Explaining the beats with a path difference. NZQA’s 2024 report says many candidates did: beats come from two slightly different frequencies drifting in and out of phase over time, wherever you stand.

How did it go?

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