Question

The fundamental frequency (in Hz) of an open pipe is 594 Hz. What is the fundamental frequency if one end is closed?

Answer 1

The smallest frequency of the stationary wave produced in an open pipe is termed as the fundamental frequency of the pipe or the first harmonic. The formation of nodes in an open pipe and a pipe closed on one side is different. Hence considering the smallest value of frequency for the two pipes and comparing them, the fundamental frequency of the pipe with one end closed can be obtained.
Formula used:
${{v}_{CP}}=\dfrac{(2n-1)V}{4L}$
${{v}_{OP}}=\dfrac{nV}{2L}$

Complete answer:
Let us say a pipe of length ‘L’ is open on either side. If V is the speed of the wave in space than in such case the frequency of ‘n ’ mode of vibration is given by,
${{v}_{OP}}=\dfrac{nV}{2L}$
Similarly let us say we close the same pipe on one of its ends. Hence the frequency of the ‘n’ modes of vibration is given by,
${{v}_{CP}}=\dfrac{(2n-1)V}{4L}$
For the fundamental mode of vibration the frequency is the smallest. Hence we can imply that n=1. Therefore we get the fundamental frequency of the open pipe as,
${{v}_{OP}}=\dfrac{V}{2L}....(1)$
Similarly the fundamental frequency of the pipe closed at one of its end is,
$\begin{aligned} & {{v}_{CP}}=\dfrac{(2(1)-1)V}{4L} \\\ & \therefore {{v}_{CP}}=\dfrac{V}{4L}....(2) \\\ \end{aligned}$
Dividing equation 1 and 2 we get,
$\begin{aligned} & \dfrac{{{v}_{OP}}}{{{v}_{CP}}}=\dfrac{\dfrac{V}{2L}}{\dfrac{V}{4L}} \\\ & \Rightarrow \dfrac{{{v}_{OP}}}{{{v}_{CP}}}=\dfrac{4}{2}=2 \\\ & \because {{v}_{OP}}=594Hz, \\\ & \Rightarrow {{v}_{CP}}=\dfrac{594Hz}{2} \\\ & \therefore {{v}_{CP}}=297Hz \\\ \end{aligned}$
Hence the fundamental frequency of the pipe closed at one of the ends is 297Hz.

Note:
It is to be noted that the first fundamental frequency also corresponds to the first harmonic which means only one node. The above equations can be obtained by analytical treatment of stationary waves in a pipe. The speed of the wave in the pipe depends on the density of the medium and the pressure.