An open pipe is in resonance in `2nd` harmonic with frequency `f_(1)`. Now one end of the tube is closed and frequency is increased to `f_(2)` such that the resonance again ocuurs in `nth` harmonic. Choose the correct option

Options

A. (a)`n=3,f_(2)=(3)/(4)f_(1)`
B. (b)`n=3,f_(2)=(5)/(3)f_(1)`
C. (c)`n=5,f_(2)=(3)/(4)f_(1)`
D. (d)`n=5,f_(2)=5/4 f_(1)`

Explanation

To solve the problem, we need to analyze the frequencies of the harmonics in both the open and closed pipes.

### Step-by-Step Solution:

1. **Understanding the Open Pipe:**
- For an open pipe, the frequency of the nth harmonic is given by:
\[ f_n = \frac{n \cdot v}{2L} \]
- In this case, we are given that the pipe is in resonance in the 2nd harmonic with frequency \( f_1 \):
\[ f_1 = \frac{2 \cdot v}{2L} = \frac{v}{L} \]
- Thus, we can express the speed of sound in terms of the frequency and length:
\[ v = f_1 \cdot L \]

2. **Understanding the Closed Pipe:**
- When one end of the pipe is closed, the frequency of the nth harmonic is given by:
\[ f_n = \frac{(2n-1) \cdot v}{4L} \]
- We denote the new frequency as \( f_2 \) when the pipe is closed and resonates in the nth harmonic:
\[ f_2 = \frac{(2n-1) \cdot v}{4L} \]

3. **Relating \( f_1 \) and \( f_2 \):**
- We can substitute \( v \) from the open pipe equation into the closed pipe equation:
\[ f_2 = \frac{(2n-1) \cdot (f_1 \cdot L)}{4L} \]
- Simplifying this gives:
\[ f_2 = \frac{(2n-1) \cdot f_1}{4} \]

4. **Finding the Relationship Between Frequencies:**
- Rearranging the equation gives us:
\[ f_2 = \frac{(2n-1)}{4} f_1 \]
- This shows how \( f_2 \) is related to \( f_1 \) based on the harmonic number \( n \).

5. **Choosing the Correct Option:**
- We can now analyze the options provided in the question. We need to find the correct relationship based on the derived equation:
- If \( n = 3 \), then:
\[ f_2 = \frac{(2 \cdot 3 - 1)}{4} f_1 = \frac{5}{4} f_1 \]
- If \( n = 5 \), then:
\[ f_2 = \frac{(2 \cdot 5 - 1)}{4} f_1 = \frac{9}{4} f_1 \]
- The correct option based on the derived equations is option D, which corresponds to \( f_2 = \frac{5}{4} f_1 \) when \( n = 3 \).

### Final Answer:
The correct option is **D**.