Answer to Question #88571 in Classical Mechanics for Shivam Nishad

Question

Show that only odd integral multiples of
fundamental frequency are excited in a
closed pipe.

Answer 1

For a closed pipe the node is at closed end and anti-nodes at open end.

Let, length of the closed pipe is l.

For the fundamental mode ,corresponding wavelength is given by,$\lambda_1 = 4l$

Now, if $n_1$ ​ is fundamental frequency and v is velocity of sound in air then, $n_1=\dfrac{v}{\lambda_1} = \dfrac{v}{4l}$

Again for first overtone ,$\lambda_3 = \dfrac{4l}{3}$

So, frequency $n_3 = \dfrac{v}{\lambda_3} = \dfrac{3v}{4l} = 3n_1$ [As, $n_1 = \dfrac{v}{4l}$ ]

 

Similarly, $n_5 = 5n_1$

 

So, frequency of the m-th overtone is (2m +1) $n_1$ ​ . [where, $n_1$ ​ is the fundamental frequency]

 

So it is shows that in a closed pipe only odd harmonics are produced.

 

Answer: In a closed pipe odd harmonics are produced.