A pipe which is open at are end and closed at the other end is called a closed pipe.
Consider a closed pipe of length l. Let v be the velocity of sound in air. The air column in a closed pipe vibrates in such. a way that always antinodes formed at the open end and node is formed at the closed end. Let f1, f2,and f3 be the frequencies and l1, l2 art l3 be the wavelengths of 1st, 2nd and 3rd modes of vibration respectively.
For 1st mode (fundamental mode) l = λ1/4 or λ1= 4l but
For i 2nd mode (1st overtone)
from (1)
For 3rd mode (2nd overtone)
from (1)
From (1), (2) and (3)
f1: f2: f3 = 1:2:3
Therefore in the case of an closed pipe, the frequencies of overtones are odd harmonics of the fundamental.
