The rope shown at an instant is carrying a wave travelling towards right, created by a source vibrating at a frequency $n$. Consider the following statements

$I.$ The speed of the wave is $4n \times ab$

$II.$ The medium at $a$ will be in the same phase as $d$ after $\frac{4}{{3n}}s$

$III.$ The phase difference between $b$ and $e$ is $\frac{{3\pi }}{2}$

Which of these statements are correct

Four wires of identical length, diameters and of the same material are stretched on a sonometre wire. If the ratio of their tensions is $1 : 4 : 9 : 16$ then the ratio of their fundamental frequencies are

A transverse wave of frequency $500 \,Hz$ and speed $100 \,m / s$ is travelling in the positive $x$-direction on a long string. At time $t=0 \,s$, the displacements at $x=0.0 \,m$ and at $x=0.25 \,m$ are $0.0 \,m$ and $0.02 \,m$, respectively. The displacement at $x=0.2 \,m$ at $t=5 \times 10^{-4} s$ is ………… $m$

Fundamental frequency of sonometer wire is $n$. If the length, tension and diameter of wire are tripled, the new fundamental frequency is

A sonometer wire of resonating length $90 \mathrm{~cm}$ has a fundamental frequency of $400 \mathrm{~Hz}$ when kept under some tension. The resonating length of the wire with fundamental frequency of $600 \mathrm{~Hz}$ under same tension________.$\mathrm{cm}$.