The figure below shows the shape of a part of a long string in which transverse waves are produced by attaching one end of string to a tuning fork of frequency $250 \mathrm{Hz}.$ What is the velocity of the waves?

A steel wire of length $1 \mathrm{m}$, mass $0.1 \mathrm{kg}$ and having a uniform cross-sectional area of ${10}^{-6} {\mathrm{m}}^2$, is rigidly fixed at both the ends. The temperature of the wire is lowered by $20 °\mathrm{C}$. If the wire is vibrating in fundamental mode, find the frequency $($in $\mathrm{Hz})$.

$(Y_{steel}=2\times {10}^{11} \mathrm{N} {\mathrm{m}}^{-2}, {\alpha }_{steel}=1.21\times {10}^{-5} °{\mathrm{C}}^{-1})$

A progressive wave on a string having linear mass density $\rho$ is represented by $y=A\sin \left(\frac{2\mathrm{\pi }}{{\displaystyle \lambda }}x-\omega t\right),$ where  $y$ is in $\mathrm{mm}$, $t$ is on $\mathrm{s}$ and $x$ is in $\mathrm{cm}$. Find the total energy (in $\mathrm{\mu J}$) passing through origin from $t=0 \mathrm{s}$ to  $t=\frac{\mathrm{\pi }}{2{\displaystyle \omega }} \mathrm{s}$.

[Take $\rho =3\times {10}^{-2} \mathrm{kg} {\mathrm{m}}^{-1}, A=1 \mathrm{mm}, \omega =100 \mathrm{rad} {\mathrm{s}}^{-1}, \lambda =16 \mathrm{cm}$]