An open and a closed pipe have same length ratio of frequencies of their nth overtone is

Options

A. `(n+1)/(2n+1)`
B. `(2(n+1))/(2n+1)`
C. `(n)/(2n+1)`
D. `(n+1)/(2n)`

Answer

To find the ratio of frequencies of the nth overtone of an open pipe and a closed pipe, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Frequencies of the Pipes**:
- For an open pipe, the frequency of the nth overtone is given by the formula:
\[ f_n^{\text{open}} = (n + 1) \frac{V}{2L} \]
- For a closed pipe, the frequency of the nth overtone is given by the formula:
\[ f_n^{\text{closed}} = (2n + 1) \frac{V}{4L} \]

2. **Set Up the Ratio**:
- We need to find the ratio of the frequency of the open pipe to the frequency of the closed pipe:
\[ \text{Ratio} = \frac{f_n^{\text{open}}}{f_n^{\text{closed}}} \]

3. **Substitute the Formulas**:
- Substitute the expressions for the frequencies into the ratio:
\[ \text{Ratio} = \frac{(n + 1) \frac{V}{2L}}{(2n + 1) \frac{V}{4L}} \]

4. **Simplify the Expression**:
- The \(V\) and \(L\) cancel out:
\[ \text{Ratio} = \frac{(n + 1) \cdot 4}{(2n + 1) \cdot 2} \]
- Simplifying further gives:
\[ \text{Ratio} = \frac{4(n + 1)}{2(2n + 1)} = \frac{2(n + 1)}{2n + 1} \]

5. **Final Result**:
- Therefore, the ratio of the frequencies of the nth overtone of the open pipe to the closed pipe is:
\[ \text{Ratio} = \frac{2(n + 1)}{2n + 1} \]

### Conclusion:
The ratio of frequencies of the nth overtone of an open pipe to that of a closed pipe is:
\[ \frac{2(n + 1)}{2n + 1} \]