Following are equations of four waves :

(i) $y_1=a\sin \mathrm{\omega }\left(t-\frac{\mathit{x}}{\mathit{\nu }}\right)$ (ii) $y_2 =a\cos \mathrm{\omega }\left(t+\frac{\mathit{x}}{\mathit{\nu }}\right)$ (iii) $z_1 =a\sin \mathrm{\omega }\left(t-\frac{\mathit{x}}{\mathit{\nu }}\right)$ (iv) $z_2 =a\cos \mathrm{\omega }\left(t+\frac{\mathit{x}}{\mathit{\nu }}\right)$

Which of the following statements is/are correct?

The vibrations of a string of length $600 \mathrm{cm}$ fixed at both ends are represented by the equation

$y= 4 \sin \left(\mathrm{\pi } \frac{x}{15}\right) \cos (96 \mathrm{\pi }t)$

where $x$ and $y$ are in $\mathrm{cm}$ and $t$ in seconds.

A massless rod $BD$ is suspended by two identical massless strings $AB$ and $CD$ of equal lengths. A block of mass $‘m’$ is suspended point $P$ such that $BP$ is equal to $x$, if the fundamental frequency of the left wire is twice the fundamental frequency of right wire, then the value of $x$ is:

A transverse sinusoidal wave moves along a string in the positive $x$ -direction at a speed of $10 \mathrm{cm} {\mathrm{s}}^{-1}$. The wavelength of the wave is $0.5 \mathrm{m}$ and its amplitude is $10 \mathrm{cm}.$ At a particular time $t$ the snap-shot of the wave is shown in the figure. The velocity of point $P$ when its displacement is $5 \mathrm{cm}$ is

Figure:

A $20 \mathrm{cm}$ long string, having a mass of $1.0 \mathrm{g}$, is fixed at both ends. The tension in the string is $0.5 \mathrm{N}$. The string is set into vibration using an external vibrator of frequency $100 \mathrm{Hz}$. Find the separation (in $\mathrm{cm}$) between the successive nodes on the string.