A string with constant linear mass density $\mu$ is tied to a fixed wall inside a dry room. It is stretched horizontally and passes over a frictionless pulley to hang vertically outside. A cylindrical bucket of initial mass $M$ is attached to the hanging end of the string, keeping the system taut and at rest. At time $t=0$, rain begins to fall vertically outside. The rain accumulates in the bucket at a constant mass rate $R$ (in $\text{kg/s}$). The raindrops strike the bucket with a constant downward speed $u$. Assuming the string does not stretch, what is the speed $v$ of a transverse wave pulse on the horizontal section of the string as a function of time $t > 0$?