If `v_(1) , v_(2) and v_(3)` are the fundamental frequencies of three segments of stretched string , then the fundamental frequency of the overall string is

Options

A. `v_(1) + v_(2) + v_(3)`
B. `[(1)/(v_(1)) + (1)/(v_(2)) + (1)/( v_(3))]^(-1)`
C. `v_(1) v_(2) v_(3)`
D. `[v_(1) v_(2) v_(3)]^(1//3)`

Explanation

To find the fundamental frequency of the overall string when given the fundamental frequencies of three segments of a stretched string, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Fundamental Frequency Formula:**
The fundamental frequency \( f \) of a stretched string is given by the formula:
\[ f = \frac{1}{2L} \sqrt{\frac{T}{\mu}} \]
where:
- \( L \) is the length of the string segment,
- \( T \) is the tension in the string,
- \( \mu \) is the linear mass density of the string.

2. **Write the Frequencies for Each Segment:**
For the three segments of the string, we can write the fundamental frequencies as:
- For segment 1:
\[ f_1 = \frac{1}{2L_1} \sqrt{\frac{T}{\mu_1}} \]
- For segment 2:
\[ f_2 = \frac{1}{2L_2} \sqrt{\frac{T}{\mu_2}} \]
- For segment 3:
\[ f_3 = \frac{1}{2L_3} \sqrt{\frac{T}{\mu_3}} \]

3. **Combine the Segments:**
When the segments are combined to form an overall string, the total length \( L \) of the string is:
\[ L = L_1 + L_2 + L_3 \]

4. **Express the Overall Frequency:**
The fundamental frequency \( f \) of the overall string can be expressed as:
\[ f = \frac{1}{2L} \sqrt{\frac{T}{\mu}} \]
where \( \mu \) is the effective linear mass density of the combined string.

5. **Relate the Frequencies:**
From the individual segment frequencies, we can express the lengths in terms of the frequencies:
\[ L_1 = \frac{1}{2f_1} \sqrt{\frac{T}{\mu_1}}, \quad L_2 = \frac{1}{2f_2} \sqrt{\frac{T}{\mu_2}}, \quad L_3 = \frac{1}{2f_3} \sqrt{\frac{T}{\mu_3}} \]

6. **Substitute into the Overall Frequency:**
Substitute the expressions for \( L_1, L_2, \) and \( L_3 \) into the equation for \( L \):
\[ L = \frac{1}{2f_1} \sqrt{\frac{T}{\mu_1}} + \frac{1}{2f_2} \sqrt{\frac{T}{\mu_2}} + \frac{1}{2f_3} \sqrt{\frac{T}{\mu_3}} \]

7. **Final Expression for Overall Frequency:**
The overall frequency \( f \) can be expressed in terms of the individual frequencies:
\[ \frac{1}{f} = \frac{1}{f_1} + \frac{1}{f_2} + \frac{1}{f_3} \]
Thus, the overall fundamental frequency \( f \) is given by:
\[ f = \frac{1}{\frac{1}{f_1} + \frac{1}{f_2} + \frac{1}{f_3}} \]

### Final Answer:
The fundamental frequency of the overall string is:
\[ f = \frac{1}{\frac{1}{f_1} + \frac{1}{f_2} + \frac{1}{f_3}} \]