The length of two open organ pipes are `l` and `(l+deltal)` respectively. Neglecting end correction, the frequency of beats between them will be approximately

Options

A. `(v)/(2l)`
B. `(v)/(4l)`
C. `(vDelta l)/(2l^(2))`
D. `(vDelta l)/(l)`

Explanation

To find the frequency of beats between two open organ pipes of lengths \( l \) and \( l + \Delta l \), we can follow these steps:

### Step 1: Understand the fundamental frequency of an open organ pipe
The fundamental frequency \( f \) of an open organ pipe is given by the formula:
\[ f = \frac{v}{2L} \]
where \( v \) is the speed of sound in air and \( L \) is the length of the pipe.

### Step 2: Calculate the frequency of each pipe
For the first pipe of length \( l \):
\[ f_1 = \frac{v}{2l} \]

For the second pipe of length \( l + \Delta l \):
\[ f_2 = \frac{v}{2(l + \Delta l)} \]

### Step 3: Determine the beat frequency
The beat frequency \( f_{beat} \) is the absolute difference between the two frequencies:
\[ f_{beat} = |f_1 - f_2| \]
Substituting the expressions for \( f_1 \) and \( f_2 \):
\[ f_{beat} = \left| \frac{v}{2l} - \frac{v}{2(l + \Delta l)} \right| \]

### Step 4: Simplify the expression
Factor out \( \frac{v}{2} \):
\[ f_{beat} = \frac{v}{2} \left| \frac{1}{l} - \frac{1}{l + \Delta l} \right| \]

### Step 5: Find a common denominator
The common denominator for the fractions is \( l(l + \Delta l) \):
\[ f_{beat} = \frac{v}{2} \left| \frac{(l + \Delta l) - l}{l(l + \Delta l)} \right| \]
This simplifies to:
\[ f_{beat} = \frac{v}{2} \left| \frac{\Delta l}{l(l + \Delta l)} \right| \]

### Step 6: Approximate for small \( \Delta l \)
If \( \Delta l \) is small compared to \( l \), we can approximate \( l + \Delta l \approx l \):
\[ f_{beat} \approx \frac{v}{2} \cdot \frac{\Delta l}{l^2} \]

### Final Result
Thus, the frequency of beats between the two open organ pipes is approximately:
\[ f_{beat} \approx \frac{v \Delta l}{2l^2} \]

---