The second overtone of an open organ pipe has the same frequency as the first overtone of a closed pipe `L` metre long. The length of the open pipe will be

Options

A. `(L)/(2)`
B. `4L`
C. `L`
D. `2L`

Explanation

To find the length of the open organ pipe when the second overtone of the open pipe has the same frequency as the first overtone of a closed pipe of length `L`, we can follow these steps:

### Step 1: Understand the Frequencies of the Pipes
- For a closed organ pipe, the frequency of the first overtone (which is the second harmonic) is given by:
\[ f_{\text{closed}} = \frac{3v}{4L} \]
where \( v \) is the speed of sound in air.

- For an open organ pipe, the frequency of the second overtone (which is the third harmonic) is given by:
\[ f_{\text{open}} = \frac{3v}{2L'} \]
where \( L' \) is the length of the open pipe.

### Step 2: Set the Frequencies Equal
Since the second overtone of the open pipe has the same frequency as the first overtone of the closed pipe, we can set the two frequencies equal:
\[ \frac{3v}{4L} = \frac{3v}{2L'} \]

### Step 3: Simplify the Equation
We can cancel \( 3v \) from both sides (assuming \( v \neq 0 \)):
\[ \frac{1}{4L} = \frac{1}{2L'} \]

### Step 4: Cross Multiply
Cross multiplying gives us:
\[ 2L' = 4L \]

### Step 5: Solve for \( L' \)
Dividing both sides by 2:
\[ L' = 2L \]

### Conclusion
The length of the open pipe \( L' \) is \( 2L \).