1. Third normal mode of an open pipe

For an open pipe, the allowed frequencies are $f_n = \frac{n v}{2L}, \quad n=1,2,3,\dots$

So for the third normal mode: $f_3 = \frac{3v}{2L}$

2. Use the beat condition

The tuning fork frequency is $f = 256\,\text{Hz}$

It gives 1 beat per second with the third normal mode, so the difference in frequencies is $|f_3 - 256| = 1$

Hence, $f_3 = 255\,\text{Hz} \quad \text{or} \quad 257\,\text{Hz}$

3. Find the pipe length

Using $L = \frac{3v}{2f_3}$ with $v=340\,\text{m s}^{-1}$:

• If $f_3=255$ Hz, $L = \frac{3\times 340}{2\times 255} = \frac{1020}{510} = 2.0\,\text{m} = 200\,\text{cm}$

• If $f_3=257$ Hz, $L = \frac{1020}{514} \approx 1.984\,\text{m} = 198.4\,\text{cm}$

4. Match with the given options

Among the options, the exact value obtained is $L = 200\,\text{cm}$

So the correct option is D.

5. Comparison with stored answer

Stored correct answer: D

This matches the derived answer.