I. In a closed organ pipe, longitudinal standing waves can be formed. II. In a closed organ pipe (closed at one end), only odd harmonics are present. III. The harmonics which are present in a pipe, open at both ends are odd harmonics only. Which of the following statements related to organ pipe is/are correct?

Question

I. In a closed organ pipe, longitudinal standing waves can be formed. II. In a closed organ pipe (closed at one end), only odd harmonics are present. III. The harmonics which are present in a pipe, open at both ends are odd harmonics only. Which of the following statements related to organ pipe is/are correct? (A) Only I (B) Both II and III (C) Both I and II (D) I, II and III

Tardigrade Admin · Accepted Answer

In a closed organ pipe, two waves travelling in opposite direction superimpose with each other to develop a wave pattern which is standing or stationary. Possible wavelengths of stationary waves in this type of organ pipe is given as

λ_{n} = \frac{4 L}{( 2 n + 1 )}

or λ_{n} = 4 L, \frac{4 L}{3}, \frac{4 L}{5}, \dots

As frequency,

v = \frac{v}{λ}

(On putting,

n = 0, 1, 2, 3 \dots

)
So, frequencies of

n

th harmonics can be given as,

v_{n} = \frac{v}{4 L}, \frac{3 v}{4 L}, \frac{5 v}{4 L}, \dots

∴ v_{1} : v_{2} : v_{3} \dots = 1 : 3 : 5 \dots

So, only odd harmonics are present in case of closed organ pipe.
However, in case of open organ pipe, frequencies of

n

th harmonics,

v_{n} = \frac{n v}{2 L}

Thus, even and odd or all the harmonics are present. So, statements I and II are correct but III is incorrect.