Tube `A` has both ends open while tube `B` has one closed, otherwise they are identical. The ratio of fundamental frequency of tube `A and B` is

Options

A. `1:2`
B. `2:1`
C. `1:4`
D. `4:1`

Explanation

To solve the problem of finding the ratio of the fundamental frequencies of tube A (both ends open) and tube B (one end closed), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Fundamental Frequency of Open and Closed Tubes**:
- For a tube with both ends open (tube A), the fundamental frequency \( f_A \) is given by the formula:
\[ f_A = \frac{V}{2L} \]
- For a tube with one end closed (tube B), the fundamental frequency \( f_B \) is given by the formula:
\[ f_B = \frac{V}{4L} \]
Here, \( V \) is the speed of sound in air, and \( L \) is the length of the tube.

2. **Set Up the Ratio of Frequencies**:
- We need to find the ratio \( \frac{f_A}{f_B} \):
\[ \frac{f_A}{f_B} = \frac{\frac{V}{2L}}{\frac{V}{4L}} \]

3. **Simplify the Ratio**:
- When we divide the two fractions, we can simplify:
\[ \frac{f_A}{f_B} = \frac{V}{2L} \times \frac{4L}{V} \]
- The \( V \) and \( L \) terms cancel out:
\[ \frac{f_A}{f_B} = \frac{4}{2} = 2 \]

4. **Express the Ratio**:
- Therefore, the ratio of the fundamental frequencies \( f_A \) to \( f_B \) is:
\[ \frac{f_A}{f_B} = 2:1 \]

5. **Conclusion**:
- The final answer is that the ratio of the fundamental frequency of tube A to tube B is \( 2:1 \).