The fundamental frequency of an open organ pipe is n. If one of its ends is closed then its fundamental frequency will be –

Options

A. `n/2`
B. 2n
C. n
D. 3n

Answer

To solve the problem, we need to understand the relationship between the fundamental frequency of an open organ pipe and a closed organ pipe. Here’s a step-by-step solution:

### Step 1: Understand the Fundamental Frequency of an Open Organ Pipe
- An open organ pipe has both ends open, which means it supports a standing wave with antinodes at both ends.
- The fundamental frequency (first harmonic) occurs when the length of the pipe (L) is half the wavelength (λ).

\[ L = \frac{\lambda}{2} \]

### Step 2: Relate Frequency to Wavelength
- The frequency (f) of a wave is given by the formula:

\[ f = \frac{v}{\lambda} \]

where \( v \) is the speed of sound in air.

### Step 3: Substitute for Wavelength
- From the relationship \( L = \frac{\lambda}{2} \), we can express the wavelength as:

\[ \lambda = 2L \]

- Substituting this into the frequency equation gives:

\[ f = \frac{v}{2L} \]

### Step 4: Given Frequency for Open Pipe
- We are given that the fundamental frequency for the open organ pipe is \( n \):

\[ n = \frac{v}{2L} \]

### Step 5: Understand the Fundamental Frequency of a Closed Organ Pipe
- A closed organ pipe has one end closed and one end open. The standing wave pattern has a node at the closed end and an antinode at the open end.
- The fundamental frequency occurs when the length of the pipe (L) is a quarter of the wavelength (λ):

\[ L = \frac{\lambda}{4} \]

### Step 6: Relate Frequency to Wavelength for Closed Pipe
- From the relationship \( L = \frac{\lambda}{4} \), we can express the wavelength as:

\[ \lambda = 4L \]

- Substituting this into the frequency equation gives:

\[ f' = \frac{v}{4L} \]

### Step 7: Express the New Frequency in Terms of n
- We can relate this new frequency \( f' \) to the previously established frequency \( n \):

\[ f' = \frac{v}{4L} = \frac{1}{2} \cdot \frac{v}{2L} = \frac{1}{2} n \]

### Conclusion
- Therefore, the fundamental frequency of the closed organ pipe is:

\[ f' = \frac{n}{2} \]

### Final Answer
The fundamental frequency of the closed organ pipe will be \( \frac{n}{2} \).

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