Fill in the blanks:
$(i)$ If the fundamental frequency of a given closed pipe is $50 \ Hz$,then the frequency for the second overtone is ...... .
$(ii)$ The speed of sound in air at $STP$ is ...... .
$(iii)$ In the case of sound waves,in order to experience beats quite clearly,the value of the beat frequency $|f_1 - f_2|$ should not be greater than ...... .

(A) $(i)$ For a closed pipe,the frequency of the $n^{th}$ harmonic is given by $f_n = (2n - 1)f_1$,where $f_1$ is the fundamental frequency.
The first overtone is the $3^{rd}$ harmonic $(n=2)$,and the second overtone is the $5^{th}$ harmonic $(n=3)$.
Therefore,the frequency of the second overtone is $f_3 = 5 \times f_1 = 5 \times 50 \ Hz = 250 \ Hz$.
$(ii)$ The speed of sound in air at $STP$ ($0^{\circ}C$ and $1 \ atm$) is approximately $332 \ m/s$.
$(iii)$ For the human ear to perceive beats clearly,the beat frequency $|f_1 - f_2|$ should generally not exceed $6 \ Hz$ to $7 \ Hz$ due to the persistence of hearing.