A vertical pipe open at both ends is partially submerged in water . A tuning fork is unknown frequency is placed near the top of the pipe and made to vibrate . The pipe can be moved up and down and thus length of air column the pipe can be adjusted. For definite lengths of air column in the pipe, standing waves will be set up as a result of superposition of sound waves travelling in opposite directions. Smallest value of length of air column , for which sound intensity is maximum is `10 cm`[ take speed of sound , `v = 344 m//s`].
Answer the following questions.
Length of air column for third resonance will be

Options

A. `30 cm`
B. `45 cm`
C. `20 cm`
D. `50 cm`

Explanation

To find the length of the air column for the third resonance in a vertical pipe open at both ends, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Resonance in a Pipe Open at Both Ends**:
- In a pipe open at both ends, the fundamental frequency (first harmonic) occurs when there is one antinode at each open end. The length of the pipe (L) corresponds to half the wavelength (λ) of the sound wave.
- For the first harmonic (fundamental frequency), the relationship is given by:
\[ L_1 = \frac{\lambda}{2} \]

2. **Identify the Relationship for Higher Harmonics**:
- For the second harmonic, the length of the pipe corresponds to one full wavelength:
\[ L_2 = \lambda \]
- For the third harmonic, the length of the pipe corresponds to one and a half wavelengths:
\[ L_3 = \frac{3\lambda}{2} \]

3. **Given Information**:
- The smallest length of the air column for maximum sound intensity (first resonance) is given as \( L_1 = 10 \, \text{cm} \).
- From the first harmonic formula, we can express the wavelength:
\[ L_1 = \frac{\lambda}{2} \implies \lambda = 2L_1 = 2 \times 10 \, \text{cm} = 20 \, \text{cm} \]

4. **Calculate the Length for Third Resonance**:
- Using the wavelength calculated, we can now find the length of the air column for the third resonance:
\[ L_3 = \frac{3\lambda}{2} = \frac{3 \times 20 \, \text{cm}}{2} = \frac{60 \, \text{cm}}{2} = 30 \, \text{cm} \]

5. **Conclusion**:
- The length of the air column for the third resonance is \( 30 \, \text{cm} \).

### Final Answer:
The length of the air column for the third resonance is **30 cm**.