Two open organ pipes of fundamental frequencies `n_(1) and n_(2)` are joined in series. The fundamental frequency of the new pipes so obtained will be

Options

A. (a)`n_(1)+n_(2)`
B. (b)`(n_(1)n_(2))/(n_(1)+n_(2))`
C. (c)`(n_(1)+n_(2))/(2)`
D. (d)`sqrt((n_(1)^(2)+n_(2)^(2)))`

Answer

To find the fundamental frequency of two open organ pipes joined in series, we can follow these steps:

### Step 1: Understand the relationship between frequency and length of the 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,
- \( L \) is the length of the pipe.

### Step 2: Define the lengths of the two pipes
Let:
- \( L_1 \) be the length of the first pipe,
- \( L_2 \) be the length of the second pipe.

### Step 3: Express the frequencies in terms of lengths
For the first pipe with fundamental frequency \( n_1 \):
\[ n_1 = \frac{V}{2L_1} \implies L_1 = \frac{V}{2n_1} \]

For the second pipe with fundamental frequency \( n_2 \):
\[ n_2 = \frac{V}{2L_2} \implies L_2 = \frac{V}{2n_2} \]

### Step 4: Find the total length when pipes are joined in series
When the two pipes are joined in series, the total length \( L \) becomes:
\[ L = L_1 + L_2 = \frac{V}{2n_1} + \frac{V}{2n_2} \]

### Step 5: Simplify the expression for total length
Combining the lengths:
\[ L = \frac{V}{2} \left( \frac{1}{n_1} + \frac{1}{n_2} \right) \]

### Step 6: Substitute back to find the new fundamental frequency
Now, substituting \( L \) back into the frequency formula:
\[ n = \frac{V}{2L} = \frac{V}{2 \left( \frac{V}{2} \left( \frac{1}{n_1} + \frac{1}{n_2} \right) \right)} \]
This simplifies to:
\[ n = \frac{1}{\frac{1}{n_1} + \frac{1}{n_2}} = \frac{n_1 n_2}{n_1 + n_2} \]

### Final Answer
The fundamental frequency of the new pipe obtained by joining the two pipes in series is:
\[ n = \frac{n_1 n_2}{n_1 + n_2} \]