The fundamental frequency of an air column in a pipe closed at one end

1.	The fundamental frequency of an air column in a pipe closed at one end is in unison with the third overtone of an open pipe. Calculate the ratio of lengths of their air columns.
Answer» Given: Fundamental frequency for closed pipe = Third overtone of open pipe. To find: Ratio of lengths of air columns in both the pipes \((\frac{l_c}{l_o})\) Formulae: a. Fundamental frequency of pipe closed at one end: n_c = \(\frac{v}{4l_c}\) b. Fundamental frequency of pipe open at both ends: n_o = \(\frac{v}{2l_o}\) c. Third overtone of open pipe: n₃ = 4n₀ Calculation: Fundamental frequency of closed pipe (n_c) is same as third overtone of open pipe. ∴ n_c = n₃ ∴ n_c = 4n₀ ∴ \(\frac{v}{4l_c}\) = 4[\(\frac{v}{2l_o}\)] ∴ \(\frac{v}{4l_c}\) = \(\frac{2}{l_o}\) ∴ \(\frac{l_c}{l_o}\)= \(\frac18\) The ratio of lengths of air columns in closed pipe and open pipe is 1:8.

The fundamental frequency of an air column in a pipe closed at one end is in unison with the third overtone of an open pipe. Calculate the ratio of lengths of their air columns.

Answer»

Given: Fundamental frequency for closed pipe = Third overtone of open pipe.

To find: Ratio of lengths of air columns in both the pipes \((\frac{l_c}{l_o})\)

Formulae: a. Fundamental frequency of pipe closed at one end:

n_c = \(\frac{v}{4l_c}\)

b. Fundamental frequency of pipe open at both ends:

n_o = \(\frac{v}{2l_o}\)

c. Third overtone of open pipe:

n₃ = 4n₀

Calculation: Fundamental frequency of closed pipe (n_c) is same as third overtone of open pipe.

∴ n_c = n₃

∴ n_c = 4n₀

∴ \(\frac{v}{4l_c}\) = 4[\(\frac{v}{2l_o}\)]

∴ \(\frac{v}{4l_c}\) = \(\frac{2}{l_o}\)

∴ \(\frac{l_c}{l_o}\)= \(\frac18\)

The ratio of lengths of air columns in closed pipe and open pipe is 1:8.

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