A closed and an open organ pipe have same lengths. If the ratio of frequencies of their seventh overtones is ((a-1)/a) then the value of a is _________.

Solution

Physics sir · 3 steps

A closed pipe skips the even harmonics, so its seventh overtone is far higher up the ladder than you expect.

1. Closed pipe. A closed pipe has only odd harmonics: f = v/4L, 3v/4L, 5v/4L, … The fundamental is the 1st term, so the seventh overtone is the 8th allowed term, the 15th harmonic: f_c = 15v/4L.
2. Open pipe. An open pipe has every harmonic: f = v/2L, 2v/2L, 3v/2L, … Its seventh overtone is the 8th harmonic: fₒ = 8v/2L = 16v/4L.

   f_c/fₒ = (15v/4L)/(16v/4L) = 15/16 = (a-1)/a

3. So. Matching 15/16 with (a-1)/a gives a = 16.

The common slip: 14. Treating the seventh overtone as the seventh allowed frequency (13th harmonic for the closed pipe, 7th harmonic for the open pipe) gives a ratio of 13/14, so a = 14 instead of 16.

The idea to keep. The n-th overtone is the (n+1)-th allowed frequency. For an open pipe that is the (n+1)-th harmonic; for a closed pipe it is the (2n+1)-th harmonic, because only odd harmonics exist.

Answer from the NTA final answer key. Solution by PrepWiser.