A closed organ pipe has length `L`. The air in it is vibrating in third overtone with maximum amplitude a . The amplitude at distance`(L)/(7)` from closed of the pipe is a) 0 b) a c) a/2 d) data insufficient

Options

A. `0`
B. `a`
C. `(a)/(2)`
D. Date insufficient

Explanation

To solve the problem, we need to determine the amplitude of the sound wave at a distance \( \frac{L}{7} \) from the closed end of a closed organ pipe when it is vibrating in its third overtone.

### Step-by-Step Solution:

1. **Understanding the Closed Organ Pipe**:
A closed organ pipe has one end closed and the other end open. The fundamental frequency and overtones can be determined based on the length of the pipe.

2. **Wavelength Calculation**:
For a closed organ pipe, the relationship between the length \( L \) of the pipe and the wavelength \( \lambda \) is given by:
\[ L = \frac{n \lambda}{4} \]
where \( n \) is the harmonic number. For the third overtone, \( n = 7 \) (since the first overtone is the second harmonic, the second overtone is the third harmonic, and so on). Thus, we have:
\[ L = \frac{7 \lambda}{4} \]
From this, we can find the wavelength:
\[ \lambda = \frac{4L}{7} \]

3. **Wave Number Calculation**:
The wave number \( k \) is defined as:
\[ k = \frac{2\pi}{\lambda} \]
Substituting the value of \( \lambda \):
\[ k = \frac{2\pi}{\frac{4L}{7}} = \frac{7\pi}{2L} \]

4. **Wave Equation**:
The wave equation for a standing wave can be expressed as:
\[ y(x) = A \sin(kx) \]
where \( A \) is the maximum amplitude. For the third overtone, we substitute \( k \):
\[ y(x) = A \sin\left(\frac{7\pi}{2L} x\right) \]

5. **Amplitude at \( \frac{L}{7} \)**:
We need to find the amplitude at a distance \( x = \frac{L}{7} \):
\[ y\left(\frac{L}{7}\right) = A \sin\left(\frac{7\pi}{2L} \cdot \frac{L}{7}\right) \]
Simplifying this:
\[ y\left(\frac{L}{7}\right) = A \sin\left(\frac{7\pi}{2L} \cdot \frac{L}{7}\right) = A \sin\left(\frac{\pi}{2}\right) = A \]

6. **Final Answer**:
The amplitude at a distance \( \frac{L}{7} \) from the closed end of the pipe is \( A \).

Thus, the correct answer is **(b) A**.