Show that an organ pipe of length 2l open at both ends has the same fundamental frequency as another pipe of length I closed at one end.

Answer

To show that an organ pipe of length \(2L\) open at both ends has the same fundamental frequency as another pipe of length \(L\) closed at one end, we will derive the fundamental frequencies of both pipes and set them equal to each other.

### Step 1: Determine the fundamental frequency of the open pipe
For a pipe open at both ends, the fundamental frequency (\(f_1\)) is given by the formula:
\[ f_1 = \frac{V}{2L_1} \]
where:
- \(V\) is the velocity of sound in air,
- \(L_1\) is the length of the pipe.

In our case, \(L_1 = 2L\), so we can substitute this into the formula:
\[ f_1 = \frac{V}{2(2L)} = \frac{V}{4L} \]

### Step 2: Determine the fundamental frequency of the closed pipe
For a pipe closed at one end, the fundamental frequency (\(f_2\)) is given by the formula:
\[ f_2 = \frac{V}{4L_2} \]
where:
- \(L_2\) is the length of the closed pipe.

In our case, \(L_2 = L\), so we can substitute this into the formula:
\[ f_2 = \frac{V}{4L} \]

### Step 3: Set the frequencies equal to each other
Since we need to show that the fundamental frequencies are the same, we set \(f_1\) equal to \(f_2\):
\[ \frac{V}{4L} = \frac{V}{4L} \]

### Step 4: Conclusion
Since both expressions for the fundamental frequency are equal, we have shown that the organ pipe of length \(2L\) open at both ends has the same fundamental frequency as another pipe of length \(L\) closed at one end.