Question 19: Waves

Question

An open pipe is in second harmonic with frequency $f_1$. One end of the tube is closed and frequency is increased to $f_2$ and it resonates again with nth harmonic. For what value of n , the ratio of $\frac{f_1}{f_2}$ is $\frac{4}{5}$ ?

Accepted Answer

Let L be the length of the pipe and v the speed of sound.
OpenÃ¢ÂÂopen pipe (both ends open), second harmonic
Ã¢ÂÂ Fundamental: $$f_{o,1} = \frac{v}{2L}$$
Ã¢ÂÂ Second harmonic: $$f_1 = 2\,f_{o,1} = \frac{v}{L}$$
ClosedÃ¢ÂÂopen pipe (one end closed), only odd harmonics:
$$f_{c,n} = n\,\frac{v}{4L}\quad(n=1,3,5,\dots)$$
At resonance with the Ã¢ÂÂn-thÃ¢ÂÂ harmonic,
$$f_2 = f_{c,n} = n\,\frac{v}{4L}$$
Ratio
$$\frac{f_1}{f_2} \;=\;\frac{	frac{v}{L}}{\,n\,	frac{v}{4L}\,}                     = \frac{4}{n}$$
Given $	frac{f_1}{f_2}= 	frac{4}{5}$,
$$\frac{4}{n} = \frac{4}{5}\quad\Longrightarrow\quad n = 5.$$
Answer: 5.

Answer

3

Answer

9

Answer

5

Answer

7

Waves

An open pipe is in second harmonic with frequency f_1. One end of the tube is closed and frequency is increased to f_2 and it resonates again with nth harmonic. For what value of n , the ratio of \frac{f_1}{f_2} is \frac{4}{5} ?

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