AS 91523 · 2025 paper · standing waves

Question 2(d)(i) Merit to Excellence

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

Two neighbouring strings can be played together. Played with no finger on it, string 2 gives 98.0 Hz. Strings 1 and 2 are played at the same time while a finger presses on string 1, and string 1's length is gradually reduced until a beat is heard.

Explain why beats form.

The 2025 paper, page 6 ↗ Marking schedule, page 6 ↗

Work it out

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

1 While string 1 is a little below 98.0 Hz, what do the two sound waves reaching you have in common, and what is different?
Nearly the same frequency, but not quite: string 1’s waves arrive slightly less often than string 2’s.
2 Suppose the two waves are in phase at one moment. What happens over the next few vibrations?
String 2’s wave gains a little on string 1’s every cycle, so they drift out of phase. After a while they are half a cycle apart (180° out of phase), then they drift back into phase, and so on.
3 What do you hear when they are in phase, and when they are out of phase?
In phase, they interfere constructively: compressions arrive with compressions and the sound is loud. Out of phase, they interfere destructively: a compression from one arrives with a rarefaction from the other and the sound is quiet.
4 So what does a listener hear over time?
The loudness rising and falling regularly: loud, quiet, loud. Each loud pulse is a beat, and the number of beats a second is the difference between the frequencies, f_beat = |f₁ − f₂|.

Answer The two notes have slightly different frequencies, so their waves keep drifting into and out of phase. In phase they interfere constructively and the sound is loud; 180° out of phase they interfere destructively and it is quiet. So the loudness pulses, at the difference between the two frequencies: those pulses are the beats.

What the marker wanted

Achieved One of: constructive interference gives a loud sound and destructive a quiet one; or the volume is heard pulsing.
Merit Explains how beats form: the two waves keep drifting into step and out again, so loud constructive interference alternates with quiet destructive interference.
Excellence With (ii), the complete answer: how beats form, and how the beats change as string 1 is shortened.

The schedule marks (d)(i) and (d)(ii) together as one item. Either explanation on its own reaches Merit; Excellence needs both, complete.

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

Common mistake

Explaining beats with a path difference. Both strings are the same distance from you; beats come from the two frequencies being slightly different, so the phase between the waves keeps changing in time. NZQA’s 2025 report lists path difference for beats among answers that did not reach Achieved.

How did it go?

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