Question : The length of an open organ pipe is twice the length of another closed organ pipe. The fundamental frequency of the open pipe is 100 Hz. The frequency of the third harmonic of the closed pipe is:
(a) 100 Hz
(b) 150 Hz
(c) 200 Hz
(d) 300 Hz
Doubt by Radhika
Solution :
Let 𝜈 = frequency of open organ pipe
𝜈' = frequency of closed organ pipe
v = velocity of sound wave in air
L' = Length of the closed organ pipe
L = Length of the open organ pipe = 2L'
n = mode of vibration
We know,
𝜈 = nv/2L
and
𝜈'=(2n-1)v/4L'
For Fundamental Frequency of an Open Organ Pipe (n=1)
𝜈1 = (1)v/2L
100 = v/2L
100 = v/2(2L') [∵L=2L']
100 = v/4L' — (1)
Frequency of the Third Harmonic of the Closed Organ Pipe (n=2)
𝜈'2=[2(2)-1]v/4L'
𝜈'2=3v/4L'
𝜈'2=3(100) [From Equation (1)]
𝜈'2=300 Hz
Hence, (d) 300 Hz, would be the correct option.