A closed pipe resonates at its fundmental frequency of `300 Hz`. Which one of the following statements is wrong ?

Options

A. if the temperature rises, the fundamental frequency increases.
B. if the pressure rises, the fundamental frequency increases.
C. The first overtone is of frequency `900 Hz`.
D. An open pipe with the same fundamental frequency has twice the length.

Explanation

To solve the question regarding the closed pipe resonating at its fundamental frequency of 300 Hz, we will analyze each statement provided in the options to determine which one is incorrect.

### Step-by-Step Solution:

1. **Understanding the Fundamental Frequency of a Closed Pipe**:
The fundamental frequency (f) of a closed pipe can be calculated using the formula:
\[ f = \frac{V}{4L} \]
where \( V \) is the speed of sound in the medium and \( L \) is the length of the pipe.

2. **Analyzing Statement 1**:
- **Statement**: If temperature rises, the fundamental frequency increases.
- **Explanation**: The speed of sound \( V \) is related to temperature \( T \) by the formula:
\[ V = \sqrt{\gamma RT} \]
where \( \gamma \) is the adiabatic index, \( R \) is the gas constant, and \( T \) is the temperature. As temperature increases, \( V \) increases, which leads to an increase in the fundamental frequency \( f \).
- **Conclusion**: This statement is **correct**.

3. **Analyzing Statement 2**:
- **Statement**: If the pressure rises, the fundamental frequency increases.
- **Explanation**: The fundamental frequency of a closed pipe is independent of pressure. The speed of sound depends on temperature and not on pressure in an ideal gas.
- **Conclusion**: This statement is **incorrect**.

4. **Analyzing Statement 3**:
- **Statement**: The first overtone is of frequency 900 Hz.
- **Explanation**: In a closed pipe, only odd harmonics are present. The first overtone (first harmonic after the fundamental) is the third harmonic. The frequency of the first overtone can be calculated as:
\[ f_1 = 3f_0 = 3 \times 300 \text{ Hz} = 900 \text{ Hz} \]
- **Conclusion**: This statement is **correct**.

5. **Analyzing Statement 4**:
- **Statement**: An open pipe with the same fundamental frequency has twice the length.
- **Explanation**: The fundamental frequency of an open pipe is given by:
\[ f = \frac{V}{2L} \]
For a closed pipe and an open pipe with the same fundamental frequency, we have:
\[ \frac{V}{4L} = \frac{V}{2L'} \]
Solving this gives \( L' = 2L \), meaning the open pipe must be twice the length of the closed pipe to have the same fundamental frequency.
- **Conclusion**: This statement is **correct**.

### Final Conclusion:
The statement that is wrong is **Statement 2**: "If the pressure rises, the fundamental frequency increases."