A block $\mathrm{M}$ 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 $\lambda_0$ is produced at point $O$ on the rope. The pulse takes time $T_{O A}$ to reach point $A$. If the wave pulse of wavelength $\lambda_0$ is produced at point $A$ (Pulse $2$) without disturbing the position of $M$ it takes time $T_{A 0}$ to reach point $O$. Which of the following options is/are correct?

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[$A$] The time $\mathrm{T}_{A 0}=\mathrm{T}_{\mathrm{OA}}$

[$B$] The velocities of the two pulses (Pulse $1$ and Pulse $2$) are the same at the midpoint of rope.

[$C$] The wavelength of Pulse $1$ becomes longer when it reaches point $A$.

[$D$] The velocity of any pulse along the rope is independent of its frequency and wavelength.