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

Options

A. `upsilon_(1)+upsilon_(2)`
B. `(upsilon_(1)upsilon_(2))/((upsilon_(1)+upsilon_(2))`
C. `(upsilon_(1)upsilon_(2))/(upsilon_(1)+upsilon_(2)`
D. `sqrt((upsilon_(1)^(2)+upsilon_(2)^(2)))`

Explanation

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

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

### Step 2: Define the Lengths of the Pipes
Let the lengths of the two pipes be \(L_1\) and \(L_2\). The fundamental frequencies of the two pipes can be expressed as:
\[ \nu_1 = \frac{v}{2L_1} \quad \text{and} \quad \nu_2 = \frac{v}{2L_2} \]

### Step 3: Combine the Lengths of the Pipes
When the two pipes are joined in series, the total length of the new pipe becomes:
\[ L = L_1 + L_2 \]

### Step 4: Find the Fundamental Frequency of the Combined Pipe
The fundamental frequency of the new pipe (ν) can be expressed using the total length:
\[ \nu = \frac{v}{2L} = \frac{v}{2(L_1 + L_2)} \]

### Step 5: Substitute the Lengths in Terms of Frequencies
Now, we can substitute \(L_1\) and \(L_2\) in terms of their frequencies:
\[ L_1 = \frac{v}{2\nu_1} \quad \text{and} \quad L_2 = \frac{v}{2\nu_2} \]
Substituting these into the equation for \(L\):
\[ L = \frac{v}{2\nu_1} + \frac{v}{2\nu_2} \]

### Step 6: Simplify the Expression
Now substituting \(L\) back into the frequency formula:
\[ \nu = \frac{v}{2\left(\frac{v}{2\nu_1} + \frac{v}{2\nu_2}\right)} \]
This simplifies to:
\[ \nu = \frac{v}{\frac{v}{\nu_1} + \frac{v}{\nu_2}} = \frac{1}{\frac{1}{\nu_1} + \frac{1}{\nu_2}} \]

### Step 7: Final Expression for the Fundamental Frequency
Using the formula for the sum of fractions, we can express this as:
\[ \nu = \frac{\nu_1 \nu_2}{\nu_1 + \nu_2} \]

### Conclusion
Thus, the fundamental frequency of the new pipe formed by joining the two pipes in series is:
\[ \nu = \frac{\nu_1 \nu_2}{\nu_1 + \nu_2} \]