If the fundamental frequency of a pipe closed at one is `512 H_(Z)` . The frequency of a pipe of the same dimension but open at both ends will be

Options

A. `1024 h_(Z)`
B. `512 H_(Z)`
C. `256 H_(Z)`
D. `128 H_(Z)`

Explanation

To solve the problem of finding the frequency of a pipe open at both ends given the fundamental frequency of a pipe closed at one end, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Information:**
- The fundamental frequency of the closed pipe (Fc) is given as 512 Hz.

2. **Understand the Relationship for Closed Pipes:**
- For a pipe closed at one end, the relationship between the wavelength (λ) and the length of the pipe (L) is given by:
\[ \frac{\lambda}{4} = L \]
- From this, we can derive the wavelength:
\[ \lambda = 4L \]

3. **Determine the Fundamental Frequency for Closed Pipes:**
- The fundamental frequency (Fc) is related to the speed of sound (C) and the wavelength (λ) by the formula:
\[ F_c = \frac{C}{\lambda} \]
- Substituting the expression for λ:
\[ F_c = \frac{C}{4L} \]

4. **Understand the Relationship for Open Pipes:**
- For a pipe open at both ends, the relationship between the wavelength (λ) and the length of the pipe (L) is given by:
\[ \frac{\lambda}{2} = L \]
- From this, we can derive the wavelength for the open pipe:
\[ \lambda = 2L \]

5. **Determine the Fundamental Frequency for Open Pipes:**
- The fundamental frequency (Fo) for an open pipe is given by:
\[ F_o = \frac{C}{\lambda} \]
- Substituting the expression for λ:
\[ F_o = \frac{C}{2L} \]

6. **Relate the Frequencies of Closed and Open Pipes:**
- We can express the fundamental frequency of the open pipe in terms of the closed pipe:
\[ F_o = 2 \times F_c \]

7. **Calculate the Fundamental Frequency for the Open Pipe:**
- Now substituting the value of Fc:
\[ F_o = 2 \times 512 \text{ Hz} = 1024 \text{ Hz} \]

8. **Conclusion:**
- Therefore, the fundamental frequency of the pipe open at both ends is **1024 Hz**.

### Final Answer:
The frequency of a pipe of the same dimension but open at both ends is **1024 Hz**.

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