Question 238: Waves

Question

An air column is of length 17 cm . The ratio of frequencies of $5^{	ext {th }}$ overtone if the air column is closed at one end to that open at both ends is (velocity of sound in air $=340 \mathrm{~ms}^{-1}$ )

Accepted Answer

Closed Pipe (One End Closed):
When an air column is closed at one end, it produces only odd harmonics. This means only certain frequencies are possible.
The formula for the $ n $th overtone in a closed pipe is that it is actually the $ (2n+1) $th harmonic.
The 5
th
overtone in this case is the 11
th
harmonic. (Because $ 2 	imes 5 + 1 = 11 $)
The frequency formula for a closed pipe is: $$ f_{	ext{closed}} = \frac{11V}{4L} $$
Open Pipe (Both Ends Open):
When the pipe is open at both ends, it can produce all harmonics (both odd and even).
The formula for the $ n $th overtone in an open pipe is that it is the $ (n+1) $th harmonic.
The 5
th
overtone here is the 6
th
harmonic. (Because $ 5+1 = 6 $)
The frequency formula for an open pipe is: $$ f_{	ext{open}} = \frac{6V}{2L} $$
Finding the Ratio:
To get the answer, divide the formula for closed pipe frequency by the formula for open pipe frequency:
$$  \frac{f_{	ext{closed}}}{f_{	ext{open}}} = \frac{11V}{4L} 	imes \frac{2L}{6V} = \frac{11 	imes 2}{4 	imes 6} = \frac{22}{24} = \frac{11}{12} $$
So, the ratio of frequencies is $ \frac{11}{12} $.

Answer

$\frac{9}{11}$

Answer

$\frac{5}{7}$

Answer

$\frac{11}{12}$

Answer

$\frac{13}{9}$

Waves

An air column is of length 17 cm . The ratio of frequencies of 5^{\text {th }} overtone if the air column is closed at one end to that open at both ends is (velocity of sound in air =340 \mathrm{~ms}^{-1} )

Choose the correct answer: