Question

Difference in frequencies between $3^{\text {rd }}$ overtone of closed pipe and $5^{\text {th }}$ harmonic of the same pipe is $400\, Hz$. Further, $3^{\text {rd }}$ harmonic of this closed pipe is equal to $6^{\text {th }}$ harmonic of another organ pipe.
If speed of sound is $330\, m/s.$ The lengths of closed pipe and open pipe are :

• A. 0.4125 m, 1.65 m
• B. 3.3 m, 1.65 m
• C. 0.825 m, 0.825 m
• D. 1.65 m, 0.825 m

Answer 1

Answer: (A)

For closed pipe,
$3^{ rd }$ overtone $=7^{\text {th }}$ harmonic $=n_{7}=\frac{7 v}{4 l_{c}}$
$5^{\text {th }}$ harmonic $n_{5}=\frac{5 v}{4 l_{c}}$
$n_{7}-n_{5}=\frac{2 v}{4 l_{c}}=400\, Hz$
$\Rightarrow n_{0}=\frac{v}{4 l_{c}}=200\, Hz$
(fundamental frequency of closed pipe)
Now $3^{\text {rd }}$ harmonic of closed pipe is equal to $6^{\text {th }}$ harmonic of open pipe
$\Rightarrow \frac{v}{4 l_{c}}=\frac{6 v}{2 l_{0}}$
$\Rightarrow \frac{v}{l_{0}}=\frac{v}{4 l_{c}}=200\, Hz$
Fundamental frequency of open pipe $=\frac{v}{2 l_{0}}=100\, Hz$
Further, $l_{ c }=\frac{330}{4 \times 200}=0.4125\, m$
$l_{0}=\frac{330}{2 \times 100}=1.65\, m$