Twang!

Classical Mechanics

A weight of mass M is hanging from a string. The string has length $\ell$ ; its mass m is uniformly distributed.

A small disturbance of the hanging mass M causes a transverse pulse wave to travel up the string. The time it takes for this pulse to reach the top of the string can be written as $t = c\sqrt{\frac{\ell}{g}},$ where c is a dimensionless number. What is the correct expression for c?