Third overtone of a closed organ pipe is in unison with fourth harmonic of an open organ pipe. Find the ratio of lengths of the two pipes.

Options

A. `(7)/(8)`
B. `(4)/(5)`
C. `(7)/(10)`
D. `(4)/(10)`

Answer

To solve the problem of finding the ratio of lengths of a closed organ pipe and an open organ pipe when the third overtone of the closed pipe is in unison with the fourth harmonic of the open pipe, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Frequencies of the Closed Organ Pipe:**
- For a closed organ pipe, the frequencies are given by:
- Fundamental frequency: \( f_1 = \frac{V}{4L_1} \)
- First overtone (3rd harmonic): \( f_2 = \frac{3V}{4L_1} \)
- Second overtone (5th harmonic): \( f_3 = \frac{5V}{4L_1} \)
- Third overtone (7th harmonic): \( f_4 = \frac{7V}{4L_1} \)
- Therefore, the frequency of the third overtone of the closed organ pipe is \( f_{closed} = \frac{7V}{4L_1} \).

2. **Identify the Frequencies of the Open Organ Pipe:**
- For an open organ pipe, the frequencies are given by:
- Fundamental frequency: \( f_1 = \frac{V}{2L_2} \)
- First harmonic: \( f_2 = \frac{2V}{2L_2} \)
- Second harmonic: \( f_3 = \frac{3V}{2L_2} \)
- Third harmonic: \( f_4 = \frac{4V}{2L_2} \)
- Therefore, the frequency of the fourth harmonic of the open organ pipe is \( f_{open} = \frac{4V}{2L_2} = \frac{2V}{L_2} \).

3. **Set the Frequencies Equal:**
- According to the problem, the third overtone of the closed organ pipe is in unison with the fourth harmonic of the open organ pipe:
\[ \frac{7V}{4L_1} = \frac{2V}{L_2} \]

4. **Eliminate \( V \) from the Equation:**
- We can cancel \( V \) from both sides of the equation:
\[ \frac{7}{4L_1} = \frac{2}{L_2} \]

5. **Cross-Multiply to Solve for the Ratio:**
- Cross-multiplying gives:
\[ 7L_2 = 8L_1 \]

6. **Express the Ratio of Lengths:**
- Rearranging the equation gives:
\[ \frac{L_1}{L_2} = \frac{7}{8} \]

7. **Final Ratio:**
- Therefore, the ratio of the lengths of the closed organ pipe to the open organ pipe is:
\[ L_1 : L_2 = 7 : 8 \]

### Conclusion:
The ratio of lengths of the closed organ pipe to the open organ pipe is \( 7:8 \).