The pulse shown in figure has a speed of $5 \mathrm{cm} {\mathrm{s}}^{-1}.$ If the linear mass density of the right string is $0.5$ that of the left string, find the ratio of the height of the transmitted pulse to that of the incident pulse.

Two coherent narrow slits emitting sound of wavelength $\lambda$ in the same phase, are placed parallel to each other at a small separation of $2\lambda$. The sound is detected by moving a detector on the screen $\mathrm{\Sigma }$ at a distance $D (>>\lambda )$ from the slit $S_1$, as shown in the figure. Find the distance $x$ such that the intensity at $P$ is equal to the intensity at $O$.

$\left(\mathrm{a}\right)$ The frequency of the sound emitted by the source and, 

$\left(\mathrm{b}\right)$ The relative amplitudes of the two waves arriving at the detector. The velocity of sound in air $=340 \mathrm{m} {\mathrm{s}}^{-1}.$

A $2.0 \mathrm{m}$-long wire having a mass of $0.10 \mathrm{kg}$ is fixed at both ends. The tension in the wire is maintained at $20.0 \mathrm{N}.$ What are the frequencies of the first three allowed modes of vibration?