A closed organ pipe of length 30 cm and an open organ pipe of length L have the same radius of 2.5 cm . The frequency of the first overtone of the closed pipe is equal to the fundamental frequency of the open pipe. If the end correction is 0.6 times the radius, the value of L is ________ cm .

Step-by-step solution

The end correction for a pipe of radius r is: e = 0.6r = 0.6 2.5 = 1.5 cm For the closed organ pipe, the end correction is applied at one end. Its effective length is: L_ c ' = L_c + e = 30 + 1.5 = 31.5 cm The first overtone of a closed pipe corresponds to the 3^ rd harmonic. Its frequency is: f_c = 3v 4L_ c ' = 3v 4 31.5 = 3v 126 = v 42 For the open organ pipe, the end correction is applied at both ends. Its effective length is: L_ o ' = L + 2e = L + 2(1.5) = L + 3 The fundamental frequency of the open pipe is: f_