Question

Show that only odd harmonics are present as overtones in the case of an air column vibrating in a pipe closed at one end.

Answer 1

Let's consider a organ pipe closed at one end and open at other end. if a vibrating of tunning fork of frequency held near its open end , it sends longitudinal waves in the pipe , they are reflected from closed end. here the incident and reflected waves interfere and formed stationary wave.
Let length of closed organ pipe is L
wave length of Longitudinal wave is $\lambda$

see figure, length of organ pipe , l = 1/4 × wavelength = $\frac{\lambda}{4}$
so, fundamental frequency = v/4l , where v is speed of sound.

first overtones , length of organ pipe ,l= $\frac{\lambda}{2}+\frac{\lambda}{4}$
l = $\frac{3\lambda}{4}$
now, frequency of first overtone = 3v/4L
e.g., 1st overtones = 3rd harmonic

similarly, we get 2nd overtone = 5th harmonic
so, nth overtone = (2n + 1)th harmonic

hence ,it is clear that only odd harmonics are present as overtones in the case of an air column vibrating in a pipe closed at one end.