Q4. A string gives 5 beats per second with a 300 Hz fork. When the string is tightened a little without crossing the reference frequency, the beats become faster. What was the frequency of the string?
It was 300 ± 5 Hz. Tightening raises its frequency; the beats got faster, so it moved further from 300 Hz, which means it was already above: 305 Hz.
Q5. Two sources produce sound whose loudness peaks every 0.25 s. What is the difference between their frequencies?
The beat frequency is 1/0.25 = 4 Hz, so the frequencies differ by 4 Hz.
Q6. Fork A gives 20 beats in 5 s with a 256 Hz fork. What are the possible frequencies of A?
The beat frequency is 20/5 = 4 Hz, so A is either 252 Hz or 260 Hz. A further test (such as loading A with wax) is needed to decide.
Q7. Two sounds s1 = a sin 400πt and s2 = a sin 404πt (t in s) are heard together. How many beats are heard per second?
ν = ω/2π gives 200 Hz and 202 Hz, so 202 − 200 = 2 beats per second (not 4, and not 1).
Q8. A fork gives 4 beats per second with a 256 Hz fork. After some wax is put on it, it still gives 4 beats per second. What was its original frequency?
It was 252 or 260 Hz. Wax lowers it. From 252 Hz the beats would increase, but from 260 Hz it can drop to 252 Hz, again 4 Hz from 256 Hz. So 260 Hz.