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  4. A pipe closed at one end produces a fundamental note of 412 Hz. It is cut into two pieces of equal length the fundamental notes produced by the two pieces are

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Q. A pipe closed at one end produces a fundamental note of $412 \,Hz$. It is cut into two pieces of equal length the fundamental notes produced by the two pieces are

Bihar CECEBihar CECE 2007

Solution:

When the dosed pipe is cut one open and one closed pipe are formed.
When pipe is closed at one end $n=\frac{v}{4 l}$
image
Given, $n=412 \,H z$
$412=\frac{v}{4 l} \,\,\,...(i)$
When pipe is cut into two equal halves then length of each is $\frac{l}{2}$.
$n_{1}=\frac{v}{4 l}$ (closed pipe)
$n_{2}=\frac{v}{2 l}$ (open pipe)
Where $l=\frac{l}{2}$
$\therefore n_{1}=\frac{v}{4\left(\frac{l}{2}\right)}$
Putting $v=1648 l$ from Eq. (i), we get
$n_{1}=\frac{1648 l}{2 l}=824 \,H z$
$n_{2}=\frac{v}{2\left(\frac{l}{2}\right)}=\frac{v}{l}=\frac{1648 l}{l}=1648\, H z$
Note: A closed pipe produces only odd harmonics while an open pipe produces both even and odd harmonics.