An ant with mass $m$ is standing peacefully on top of a horizontal, stretched rope. The rope has mass per unit length $\mathrm{\mu }$ and is under tension $F.$ Without warning, a student starts a sinusoidal transverse wave of wavelength $\mathrm{\lambda }$ propagating along the rope. The motion of the rope is in a vertical plane. What minimum wave amplitude (in $\mathrm{mm}$) will make the ant feel weightless momentarily? Assume that $m$ is so small that the presence of the ant has no effect on the propagation of the wave.
[Given: $\lambda =0.5 \mathrm{m}, \mu =0.1 \mathrm{kg}/\mathrm{m}, F=3.125 \mathrm{N},$ take $\left.g={\pi }^2\right]$

The transverse displacement $\mathrm{y}\left(\mathrm{x},\mathrm{t}\right)$ of a wave on a string is given by $\mathrm{y}\left(\mathrm{x},\mathrm{t}\right)={\mathrm{e}}^{-\left({\mathrm{ax}}^2+{\mathrm{bt}}^2+2\sqrt{\mathrm{ab}}\mathrm{xt}\right)}.$ This represents a:

A block $M$ hangs vertically at the bottom end of a uniform rope of constant mass per unit length. The top end of the rope is attached to a fixed rigid support at $O$. A transverse wave pulse (Pulse $1$) of wavelength ${\mathrm{\lambda }}_0$ is produced at point $O$ on the rope. The pulse takes time $T_{\mathit{O}\mathit{A}}$ to reach point $A$. If the wave pulse of wavelength ${\mathrm{\lambda }}_0$ is produced at point $A$ (Pulse $2$) without disturbing the position of $M$ it takes time $T_{\mathit{A}\mathit{O}}$ to reach point $O$. Which of the following options is/are correct.