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A tuning fork is a small, U-shaped piece of metal that vibrates to produce a specific musical note when struck. It is used widely in music and scientific applications because of its ability to generate a pure tone of constant frequency, often ensuring accurate pitch calibration. When a tuning fork is struck, it vibrates at its natural frequency, causing nearby air particles to oscillate and produce sound waves that travel to the listener’s ear. This makes it an excellent tool for tuning musical instruments like pianos. The frequency of the tuning fork mentioned in this exercise is 264 Hz.
This means it produces sound waves that vibrate 264 times per second, which forms the baseline or reference frequency to which the piano string is compared. By listening to the sound from both the tuning fork and the piano string simultaneously, differences in frequency become evident.

The frequency difference between two sounds can be detected through what is known as beat frequency. When two sound waves of slightly different frequencies are played together, they interfere with each other. This interference causes the perceived sound to oscillate between loud and soft at a rate equal to the difference in their frequencies.
The number of beats per second heard is referred to as the beat frequency. In our exercise, the beat frequency heard is 0.5 Hz, meaning the sound intensities cycle through loud-soft transitions twice every second. This auditory effect is key in determining the frequency of the unknown sound, in this case, the piano string by offering insights on how close its frequency is to the fixed frequency of the tuning fork.
The beat frequency formula is expressed as:

• \( \text{Beat frequency} = |f_1 - f_2| \)

where \( f_1 \) and \( f_2 \) are the frequencies of the two sound sources. Here, one of them is the tuning fork with a known frequency.

The frequency of a piano string determines how it sounds when played, with higher frequencies producing higher-pitched notes. In the exercise, the piano tuner's task is to match the piano string frequency as closely as possible to the tuning fork frequency.
Using the beat frequency, we deduced two possibilities for the piano string frequency:

• If the string frequency is higher than the tuning fork, it can be calculated as:

\[ f_1 = f_2 + \text{Beat frequency} = 264 \text{ Hz} + 0.5 \text{ Hz} = 264.5 \text{ Hz}\]

• If the string frequency is lower than the tuning fork, it represents:

\[ f_1 = f_2 - \text{Beat frequency} = 264 \text{ Hz} - 0.5 \text{ Hz} = 263.5 \text{ Hz}\]These calculations reflect the two potential pitches the piano string could be tuned to, allowing for adjustment in either direction depending on whether the piano string is tuned sharp or flat relative to the tuning fork.