The two nearest harmonics of a tube closed at one end and open at other end are 220 Hz and 260 Hz. What is the fundamental frequency of the system?

Options

A. 10Hz
B. 20 Hz
C. 30 Hz
D. 40 Hz

Answer

To find the fundamental frequency of a tube closed at one end and open at the other, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Type of Harmonics**:
- Since the tube is closed at one end and open at the other, it only supports odd harmonics. The harmonics can be represented as:
- First harmonic (fundamental frequency) = \( f_1 = U_0 \)
- Third harmonic = \( f_3 = 3U_0 \)
- Fifth harmonic = \( f_5 = 5U_0 \)
- And so on.

2. **Given Frequencies**:
- The two nearest harmonics provided are 220 Hz and 260 Hz. We can denote these as:
- \( f_n = 220 \, \text{Hz} \)
- \( f_{n+2} = 260 \, \text{Hz} \)
- Here, \( f_n \) corresponds to the \( n \)-th harmonic and \( f_{n+2} \) corresponds to the \( (n+2) \)-th harmonic.

3. **Calculate the Difference Between the Harmonics**:
- The difference between two consecutive odd harmonics is given by:
\[ f_{n+2} - f_n = 2U_0 \]
- Substituting the values:
\[ 260 \, \text{Hz} - 220 \, \text{Hz} = 2U_0 \]
- This simplifies to:
\[ 40 \, \text{Hz} = 2U_0 \]

4. **Solve for the Fundamental Frequency**:
- To find \( U_0 \):
\[ U_0 = \frac{40 \, \text{Hz}}{2} = 20 \, \text{Hz} \]

5. **Conclusion**:
- The fundamental frequency of the system is:
\[ U_0 = 20 \, \text{Hz} \]

### Final Answer:
The fundamental frequency of the system is **20 Hz**.

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