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Step-by-Step Solution
Step 1: Express the fundamental frequency of each segment
The fundamental frequency of a string of length $L$ (under uniform tension and linear mass density) is given by
$ n = \frac{v}{2L}, $
where $v$ is the wave speed on the string. If the original string is divided into three segments with lengths $L_1$, $L_2$, and $L_3$, then their fundamental frequencies $n_1$, $n_2$, and $n_3$ are
$ n_1 = \frac{v}{2L_1}, \quad n_2 = \frac{v}{2L_2}, \quad n_3 = \frac{v}{2L_3}. $
Step 2: Relate the total length to the segment lengths
The total length of the original string is the sum of the three segments:
$ L = L_1 + L_2 + L_3. $
Therefore, the fundamental frequency of the entire string of length $L$ is
$ n = \frac{v}{2L}. $
Step 3: Invert the frequencies
Taking reciprocals gives:
$ \frac{1}{n} = \frac{2L}{v}, \quad \frac{1}{n_1} = \frac{2L_1}{v}, \quad \frac{1}{n_2} = \frac{2L_2}{v}, \quad \frac{1}{n_3} = \frac{2L_3}{v}. $
Step 4: Summation of reciprocals
Notice that
$ \frac{1}{n_1} + \frac{1}{n_2} + \frac{1}{n_3} = \frac{2L_1}{v} + \frac{2L_2}{v} + \frac{2L_3}{v} = \frac{2(L_1 + L_2 + L_3)}{v} = \frac{2L}{v}. $
From this, we see that
$ \frac{1}{n} = \frac{2L}{v} = \frac{2L_1}{v} + \frac{2L_2}{v} + \frac{2L_3}{v} = \frac{1}{n_1} + \frac{1}{n_2} + \frac{1}{n_3}. $
Step 5: Final conclusion
Thus, the required relation between the original fundamental frequency $n$ and the fundamental frequencies $n_1$, $n_2$, and $n_3$ of the three segments is:
$ \displaystyle \frac{1}{n} = \frac{1}{n_1} + \frac{1}{n_2} + \frac{1}{n_3}. $
