A stationary tuning fork is in resonance with an air column in a pipe. If the tuning fork is moved with a speed of $$2 ms^{−1}$$ in front of the open end of the pipe and parallel to it, the length of the pipe should be changed for the resonance to occur with the moving tuning fork. If the speed of sound in air is $$320 ms^{−1}$$, the smallest value of the percentage change required in the length of the pipe is
____________.

Let the original frequency of the stationary tuning fork be $$f$$. The pipe is already in resonance with this fork, so for the same mode of vibration,

$$f = k\;\frac{v}{L_0}$$ where $$v = 320\;{\rm m\,s^{-1}}$$ is the speed of sound, $$L_0$$ is the original length of the air column and $$k$$ is a constant that depends on the mode (it will cancel out later).

When the tuning fork starts moving with speed $$u = 2\;{\rm m\,s^{-1}}$$ parallel to the axis of the pipe, the air column (observer) receives a Doppler-shifted frequency. For a moving source and a stationary observer,

$$f' = f\,\frac{v}{v \mp u}$$ • use $$v-u$$ when the source moves towards the pipe (frequency increases)
• use $$v+u$$ when it moves away (frequency decreases)

The pipe length is now adjusted to $$L'$$ so that the same mode of vibration (same $$k$$) resonates with the shifted frequency:

$$f' = k\,\frac{v}{L'} \quad\Rightarrow\quad L' = k\,\frac{v}{f'}$$

Divide the two length relations:

$$\frac{L'}{L_0} = \frac{f}{f'}$$

Case 1: Fork moving towards the pipe $$\frac{f}{f'} = \frac{v-u}{v} = 1 - \frac{u}{v}$$

Case 2: Fork moving away from the pipe $$\frac{f}{f'} = \frac{v+u}{v} = 1 + \frac{u}{v}$$

In either case, the magnitude of the fractional change in length is

$$\left|\frac{L' - L_0}{L_0}\right| = \left|1 - \frac{f}{f'}\right| = \frac{u}{v}$$

Substituting $$u = 2\;{\rm m\,s^{-1}}$$ and $$v = 320\;{\rm m\,s^{-1}}$$:

$$\frac{u}{v} = \frac{2}{320} = 0.00625$$

Percentage change required:

$$0.00625 \times 100 = 0.625\%$$

Hence, the smallest value of the percentage change that must be made in the length of the pipe to regain resonance with the moving tuning fork is

0.625 %