Question

The fundamental node produced by a closed organ pipe is of frequency $\nu$. The fundamental node produced by an open organ pipe of same length will be of frequency:
A. $\dfrac{\nu }{2}$​
B. $\nu$
C. 2$\nu$
D. 4$\nu$

Answer 1

When two longitudinal waves travel through the medium in opposite directions of the same frequency producing the standing waves. Because of the destructive interference the position on the standing waves can stay in a fixed position which produces the nodes. By using the formula of fundamental frequency of both the pipes and take a division then we can solve this above question.

Complete step by step solution:
Let us consider the length of the pipe is taken as L,
By using the formula of fundamental frequency of closed pipe is taken as,
${\nu _c} = \dfrac{\nu }{{4L}}$…….. (1)
Where $\nu$ is the speed of sound in air and ${\nu _c}$ is the fundamental frequency of closed pipe.
By using the fundamental frequency of open pipe and we are taking the same length is
${\nu _0} = \dfrac{\nu }{{2L}}$……. (2)
Divide equation 2 by 1
$\dfrac{{{\nu _0}}}{{{\nu _c}}} = \dfrac{{\dfrac{\nu }{{2L}}}}{{\dfrac{\nu }{{4L}}}} \\\ \implies \dfrac{{{\nu _0}}}{{{\nu _c}}} = 2 \\\ \implies {\nu _0} = 2{\nu _c} \\\ \implies {\nu _0} = 2\nu \\\$
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Therefore, the correct answer is “Option C”.

Additional Information:
Each natural frequency that an object or instrument produces has its own characteristic vibrational mode or standing wave pattern. These types of patterns are created within the instrument at certain frequencies of vibration and these specific frequencies are known as frequencies of harmonics. Any frequency other than a harmonic frequency, hence it results the disturbance is irregular and which are not repeating in the medium.

Note:
Students must remember the even and odd harmonics are present in an open organ pipe.
But only odd harmonics are present in a closed organ pipe and all even harmonics are absent in a closed organ pipe.
The magnitude of any frequency produced by an organ pipe is less than fundamental frequency.