As an empty vessel in filled with water, its frequency

Options

A. increases
B. decreases
C. remain the same
D. none of the above

Answer

To solve the question regarding how the frequency of an empty vessel changes when it is filled with water, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Concept of Frequency in a Vessel:**
- The frequency of a sound wave in a vessel (like an open organ pipe) is determined by the length of the air column inside it. The fundamental frequency (f) is inversely proportional to the length (L) of the air column.
- The formula for the fundamental frequency of a sound wave in an open pipe is given by:
\[ f = \frac{1}{2L} \sqrt{\frac{\gamma RT}{M}} \]
- Here, \( \gamma \) is the adiabatic index, \( R \) is the gas constant, \( T \) is the temperature, and \( M \) is the molecular weight of the gas.

2. **Analyzing the Change When Water is Added:**
- When the vessel is empty, the length of the air column is at its maximum (let's call it L).
- When water is added, the effective length of the air column decreases (let's call this new length L').
- Since the water surface behaves as a fixed end, the new length of the air column (L') is less than the original length (L).

3. **Applying the Frequency Formula:**
- The new fundamental frequency when the vessel is filled with water can be expressed as:
\[ f' = \frac{1}{2L'} \sqrt{\frac{\gamma RT}{M}} \]
- Since L' < L, it follows that:
\[ f' > f \]
- This means that the frequency increases when the length of the air column decreases.

4. **Conclusion:**
- Therefore, as the vessel is filled with water, the frequency of the sound produced by the vessel increases.

### Final Answer:
The frequency of the vessel increases when it is filled with water.

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