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 the lengths of their air columns.
Answer» Pipe closed at one end : fundamental frequency, n_c = \(\frac v{4L_c}\) column is 51.8 cm. n_o = \(\frac v{2L_o}\) In this case, the frequency of the third over tone = \(\frac {4v}{2L_o}\) = \(\frac {2v}{L_o}\) By the data, \(\frac v{4L_c}\) = \(\frac {2v}{L_o}\) ∴ \(\frac {L_o}{L_c}\) = 8 or \(\frac {L_c}{L_o}\) = \(\frac18\)

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 the lengths of their air columns.

Answer»

Pipe closed at one end : fundamental frequency,

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

column is 51.8 cm.

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

In this case, the frequency of the third over tone = \(\frac {4v}{2L_o}\) = \(\frac {2v}{L_o}\)

By the data, \(\frac v{4L_c}\) = \(\frac {2v}{L_o}\)

∴ \(\frac {L_o}{L_c}\) = 8 or \(\frac {L_c}{L_o}\) = \(\frac18\)