1.A stretched string of length $L = 1\text{ m}$ and mass $m = 10\text{ g}$ is under tension $T = 400\text{ N}$. Find fundamental frequency and frequency of 3rd harmonic.
Sol: $\mu = \frac{0.01}{1} = 0.01\text{ kg/m}$. $v = \sqrt{\frac{T}{\mu}} = \sqrt{\frac{400}{0.01}} = 200\text{ m/s}$. Fundamental $f_1 = \frac{v}{2L} = \frac{200}{2} = 100\text{ Hz}$. 3rd Harmonic $f_3 = 3 f_1 = 300\text{ Hz}$.
2.A organ pipe closed at one end has fundamental frequency $250\text{ Hz}$. Find length of the pipe if speed of sound is $340\text{ m/s}$.
Sol: $f_1 = \frac{v}{4L} \implies L = \frac{v}{4 f_1} = \frac{340}{4 \times 250} = \frac{340}{1000} = 0.34\text{ m} = 34\text{ cm}$.
