6. I. In a closed organ pipe, standing waves can be formed. II. In a closed organ pipe (closed at one end),…

6. I. In a closed organ pipe, 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 statement(s) related to organ pipe is/are correct?
  1. (a) Only I
  2. (b) Both II and III
  3. (c) Both I and II
  4. (d) I, II and III

Solution

In a closed organ pipe two waves travelling in opposite directions superpose to give a stationary wave. Possible wavelengths: $\lambda_n=\dfrac{4L}{2n+1}$, i.e. $\lambda_n=4L,\;\dfrac{4L}{3},\;\dfrac{4L}{5},\dots$. Frequency $\nu=\dfrac{v}{\lambda}$ so $\nu_n=\dfrac{v}{4L}(2n+1)$. Thus $\nu_1:\nu_2:\nu_3:\dots=1:3:5\dots$ — only odd harmonics are present for a closed pipe. For an open pipe $\nu_n=\dfrac{nv}{2L}$, so all harmonics (even and odd) are present. Therefore Statements I and II are correct but III is incorrect.

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