It is desired to increse the fundamental resonance frequency in a tube which is closed at one end. This can be achieved by

Options

A. replacing the air in the tube by hydrogen gas
B. increasing the length of the tube
C. decreasing the length of the tube
D. opening the closed end of the tube

Explanation

To solve the problem of increasing the fundamental resonance frequency in a tube that is closed at one end, we can analyze the options given based on the principles of sound waves and resonance.

### Step-by-Step Solution:

1. **Understanding the Fundamentals of Resonance in a Closed Tube:**
- A tube closed at one end supports standing waves where the closed end is a displacement node and the open end is a displacement antinode.
- The fundamental frequency (first harmonic) of a closed tube is given by the formula:
\[ f = \frac{V}{4L} \]
where \( V \) is the speed of sound in the medium and \( L \) is the length of the tube.

2. **Analyzing the Effect of Replacing Air with Hydrogen:**
- The speed of sound in hydrogen is greater than in air due to its lower density.
- If we replace air with hydrogen, the speed of sound \( V \) increases, which leads to an increase in frequency:
\[ f' = \frac{V_{H_2}}{4L} > \frac{V_{air}}{4L} \]
- Therefore, replacing air with hydrogen will increase the fundamental resonance frequency.

3. **Considering the Effect of Changing the Length of the Tube:**
- The frequency is inversely proportional to the length of the tube:
\[ f \propto \frac{1}{L} \]
- If we decrease the length \( L \), the frequency \( f \) will increase:
\[ f' = \frac{V}{4L'} \quad \text{(where \( L' < L \))} \]
- Thus, decreasing the length of the tube will also increase the fundamental resonance frequency.

4. **Evaluating the Impact of Opening the Closed End of the Tube:**
- If the tube is opened at both ends, the fundamental frequency changes to:
\[ f = \frac{V}{2L} \]
- This is because the wavelength in this case is \( \lambda = 2L \), leading to a higher frequency compared to the closed tube scenario:
\[ f' = \frac{V}{2L} > \frac{V}{4L} \]
- Therefore, opening the closed end of the tube will also increase the fundamental resonance frequency.

5. **Conclusion:**
- To increase the fundamental resonance frequency in a tube closed at one end, we can:
- Replace air with hydrogen (increases speed of sound).
- Decrease the length of the tube.
- Open the closed end of the tube.

### Final Answer:
The fundamental resonance frequency can be increased by:
1. Replacing air in the tube with hydrogen.
2. Decreasing the length of the tube.
3. Opening the closed end of the tube.