Question

an open organ pipe is suddenly closed with the result as a second over tune of the closed pipe was found to be higher in frequency by 100 hertz then the first over tune of the original pipe. The fundamental frequency of the open pipe will be?​

Answer 1

Question:-

An open organ pipe is suddenly closed such that the second overtone of the closed pipe formed was found to be higher in frequency by 100 Hz than the first overtone of the original pipe. Find the fundamental frequency of the original open pipe.

Solution:-

★ Since a closed pipe can produce only odd harmonics, the second overtone of the closed pipe is the fifth harmonic, i.e.,

$\displaystyle\longrightarrow\sf {\nu_3'=\dfrac {5v}{4L}}$

★ Since an open pipe can produce all harmonics, the first overtone of the open pipe is the second harmonic, i.e.,

$\displaystyle\longrightarrow\sf {\nu_2=\dfrac {v}{L}}$

Note that the velocity of the wave and the length of the pipe don't change.

Given that,

$\displaystyle\longrightarrow\sf {\nu_3'-\nu_2=100}$

$\displaystyle\longrightarrow\sf {\dfrac {5v}{4L}-\dfrac {v}{L}=100}$

$\displaystyle\longrightarrow\sf {\dfrac {v}{L}=400}$

Then the fundamental frequency of the original open pipe is,

$\displaystyle\longrightarrow\sf {\nu_1=\dfrac {v}{2L}}$

$\displaystyle\longrightarrow\sf {\nu_1=\dfrac {1}{2}\times 400}$

$\displaystyle\longrightarrow\sf {\underline {\underline {\nu_1=200\ Hz}}}$