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Step 1: Understand the Beat Phenomenon
When two tuning forks of frequencies $f_1$ and $f_2$ are sounded together, the beat frequency $f_{\text{beats}}$ is given by the absolute difference of the two frequencies:
$|\,f_1 - f_2\,|$. In this problem, we are given that the beat frequency is initially $4\,\text{Hz}$ when the known fork has a frequency of $288\,\text{Hz}$. This means:
$|\,f_{\text{unknown}} - 288\,| = 4
$
Step 2: Determine the Possible Unknown Frequencies
From the beat frequency condition above, the unknown frequency can be either:
$f_{\text{unknown}} = 288 + 4 = 292\,\text{Hz}
$
or
$f_{\text{unknown}} = 288 - 4 = 284\,\text{Hz}
$
Step 3: Consider the Effect of Placing Wax
Placing a small amount of wax on the tuning fork lowers its frequency. After adding wax to the unknown fork, the observed beat frequency decreases from $4\,\text{Hz}$ to $2\,\text{Hz}$. This implies that the difference in frequencies becomes smaller, indicating that initially the unknown fork’s frequency must have been higher than $288\,\text{Hz}$ (so that lowering it brings it closer to $288\,\text{Hz}$, reducing the beat frequency). Hence, the correct initial frequency cannot be $284\,\text{Hz}$, because starting below $288\,\text{Hz}$ and then lowering further would increase the frequency difference.
Step 4: Conclude the Correct Frequency
Since the beat frequency is reduced upon adding wax, the unknown fork's original frequency must have been greater than $288\,\text{Hz}$. Therefore, the unknown tuning fork has a frequency of
$
292\,\text{Hz}.
