Question

An air column in open pipe is made to vibrate in fundamental mode and then made to vibrate such that 2 modes are formed in it. The ratio of this frequency (when two nodes are formed) and the fundamental frequency is

• A. 2
• B. 1:1
• C. 3:2
• D. 2:1

Answer 1

Answer: (D)

2:1

Answer 2

Solution:

For an open pipe, the allowed frequencies are given by

$$f_n = \frac{n\,v}{2L} \quad (n = 1, 2, 3, \ldots)$$

• In the fundamental mode (first harmonic), $n=1$:

$$f_1 = \frac{v}{2L}$$

• When “2 modes are formed” in the vibrating air column, it implies the second harmonic ($n=2$):

$$f_2 = \frac{2\,v}{2L} = \frac{v}{L}$$

The ratio of the frequency of the second harmonic to the fundamental frequency is

$$\frac{f_2}{f_1} = \frac{v/L}{v/(2L)} = 2$$

Thus, the ratio is $2:1$.