Question 457: Waves

Question

A pipe open at both ends and a pipe closed at one end have same length. The ratio of frequencies of their $P^{	ext {th }}$ overtone is

Accepted Answer

The problem involves comparing the frequencies of the $ P^{	ext{th}} $ overtone of a pipe open at both ends with one closed at one end, given they have the same length.
Calculating Frequencies
Open Pipe:
The frequency of the $ P^{	ext{th}} $ overtone for a pipe open at both ends is given by:
$   f_{	ext{open}} = (p+1) \frac{v}{2L}   $
Here $ v $ is the speed of sound and $ L $ is the length of the pipe.
Closed Pipe:
The frequency of the $ P^{	ext{th}} $ overtone for a pipe closed at one end is given by:
$   f_{	ext{closed}} = (2p+1) \frac{v}{4L}   $
Ratio of Frequencies
To find the ratio of the $ P^{	ext{th}} $ overtone frequency of the open pipe to the closed pipe, we use the expressions derived:
$ 	ext{Ratio} = \frac{f_{	ext{open}}}{f_{	ext{closed}}} = \frac{(p+1) \frac{v}{2L}}{(2p+1) \frac{v}{4L}} $
Simplifying this expression yields:
$ 	ext{Ratio} = \frac{2(p+1)}{(2p+1)} $
Therefore, the ratio of frequencies for the $ P^{	ext{th}} $ overtone of the open pipe to the closed pipe is $\frac{2(p+1)}{2p+1}$.

Answer

$\frac{p+1}{2 p}$

Answer

$\frac{p+1}{2 p+1}$

Answer

$\frac{2(p+1)}{2 p+1}$

Answer

$\frac{p}{2 p+1}$

Waves

A pipe open at both ends and a pipe closed at one end have same length. The ratio of frequencies of their P^{\text {th }} overtone is

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