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

Answer 1

Answer: (C)

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 $\lambda_{n} =\frac{4 L}{(2 n+1)} $
$ { or } \lambda_{n} =4 L, \frac{4 L}{3}, \frac{4 L}{5}, \ldots$
As frequency, $v=\frac{v}{\lambda}$
(On putting, $n=0,1,2,3 \ldots$ )
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}, \ldots $
$\therefore v_{1}: v_{2}: v_{3} \ldots =1: 3: 5 \ldots$
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.