Physics - Wave on String Question with Solution | TestHub

The fundamental frequency of a string fixed at both ends is given by $f=\frac{1}{2L}\sqrt{\frac{T}{\mu }}$. Let the initial length be $L_1=L$, tension $T_1=T$, and frequency $f_1=f_0$. So, $f_0=\frac{1}{2L}\sqrt{\frac{T}{\mu }}$.

Now, the length is doubled, $L_2=2L$. The tension is quadrupled, $T_2=4T$. The linear mass density $\mu$ remains the same. The new fundamental frequency $f_2$ is:$$f_2=\frac{1}{2(2L)}\sqrt{\frac{4T}{\mu }}=\frac{1}{4L}(2)\sqrt{\frac{T}{\mu }}=\frac{1}{2L}\sqrt{\frac{T}{\mu }}$$ Thus, $f_2=f_0$.