Waves

A stretched wire has a fundamental frequency $f$.When it is divided into $3$ segments, their fundamental frequencies are $f_1, f_2$, and $f_3$.Tension $T$ is constant.We need to find the relation among $f, f_1, f_2, f_3$.Step 1: Formula for fundamental frequency of a stretched stringFor a string under tension $T$, length $L$, and linear density $\mu$ :$f = \frac{1}{2L} \sqrt{\frac{T}{\mu}}$Step 2: Frequencies of the three partsSuppose the original length is divided into three parts of lengths $L_1, L_2, L_3$, so that$L = L_1 + L_2 + L_3$Then their fundamental frequencies are$f_1 = \frac{1}{2L_1} \sqrt{\frac{T}{\mu}}, \quad f_2 = \frac{1}{2L_2} \sqrt{\frac{T}{\mu}}, \quad f_3 = \frac{1}{2L_3} \sqrt{\frac{T}{\mu}}$Step 3: Express $L_i$ in terms of $f_i$From the above,$L_i = \frac{1}{2 f_i} \sqrt{\frac{T}{\mu}}$and $L = \frac{1}{2 f} \sqrt{\frac{T}{\mu}}$Step 4: Use $L = L_1 + L_2 + L_3$$\frac{1}{2f} \sqrt{\frac{T}{\mu}} = \frac{1}{2} \sqrt{\frac{T}{\mu}} \left( \frac{1}{f_1} + \frac{1}{f_2} + \frac{1}{f_3} \right)$Cancel the common factor $\frac{1}{2} \sqrt{\frac{T}{\mu}}$ :$\frac{1}{f} = \frac{1}{f_1} + \frac{1}{f_2} + \frac{1}{f_3}$