An open organ pipe has fundamental frequency
100 Hz. What frequency will be produced if its one
end is closed ?

Options

A. `100, 200, 300…….`
B. `50, 150, 250,…..`
C. `50, 100, 200, 300,……`
D. `50,100,150,200,….`

Answer

To solve the problem of finding the frequency produced by an open organ pipe when one end is closed, we can follow these steps:

### Step 1: Understand the fundamental frequency of an open organ pipe
An open organ pipe supports standing waves with both ends open, which means that there are antinodes at both ends. The fundamental frequency (first harmonic) corresponds to one complete wave fitting into the length of the pipe.

The relationship for the fundamental frequency \( f \) is given by:
\[ f = \frac{V}{2L} \]
where \( V \) is the speed of sound in air and \( L \) is the length of the pipe.

Given that the fundamental frequency is 100 Hz, we have:
\[ f = 100 \text{ Hz} \]

### Step 2: Relate frequency to wavelength
From the equation above, we can express the wavelength \( \lambda \) in terms of the frequency and the length of the pipe:
\[ \lambda = \frac{V}{f} = \frac{V}{100} \]

### Step 3: Consider the case when one end is closed
When one end of the organ pipe is closed, the closed end becomes a node (point of no displacement) and the open end remains an antinode (point of maximum displacement). For a closed organ pipe, the standing wave pattern is different; it supports odd harmonics only.

The fundamental frequency for a closed organ pipe is given by:
\[ f' = \frac{V}{4L} \]

### Step 4: Relate the frequencies of open and closed pipes
Since we already know that:
\[ f = \frac{V}{2L} = 100 \text{ Hz} \]
We can express the speed of sound \( V \) in terms of \( L \):
\[ V = 200L \]

Now substituting \( V \) into the equation for the closed pipe:
\[ f' = \frac{200L}{4L} = \frac{200}{4} = 50 \text{ Hz} \]

### Step 5: Conclusion
Thus, the frequency produced when one end of the open organ pipe is closed is:
\[ \boxed{50 \text{ Hz}} \]

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