AS 91523 · 2025 paper · standing waves

Question 2(a) Achieved

A cello's strings are held at each end by the bridge and the nut, 0.690 m apart, and can vibrate in different harmonics.

L	0.690 m

Draw a diagram of the 4th harmonic on the string, and label the nodes and antinodes.

The 2025 paper, page 5 ↗ Marking schedule, page 5 ↗

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 What must happen at the two ends of the string?
The bridge and the nut hold them still, so each end is a node.
2 What does the 4th harmonic look like on a string fixed at both ends?
Four loops fitted end to end, one half-wavelength each: the nth harmonic of a string has n loops. So 0.690 m = 4 × λ/2, and λ = 0.345 m.
3 Where do the nodes and antinodes go?
Five nodes: one at each end and three evenly spaced between, every 0.690 / 4 = 0.1725 m. Four antinodes, one in the middle of each loop. On the paper’s grid that is a loop every two squares.

Answer Four equal loops between the fixed ends: 5 nodes (N), including both ends, 0.1725 m apart, and 4 antinodes (A), one at the middle of each loop.

What the marker wanted

Achieved A correct diagram with the nodes and antinodes labelled.

In our words, from NZQA’s 2025 marking schedule.

Common mistake

Loops of different sizes, or a wave that does not finish on a node at both ends. NZQA’s 2025 report names both among answers that did not reach Achieved. Mark the five nodes on the line first, evenly spaced, then draw the loops between them.

How did it go?

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