Chapter 16, Problem 48AP

A rope of total mass m and length L is suspended vertically. Analysis shows that for short transverse pulses, the waves above a short distance from the free end of the rope can be represented to a good approximation by the linear wave equation discussed in Section 16.5. Show that a transverse pulse travels the length of the rope in a time interval that is given approximately by $\Delta t=2\sqrt{L/g}$. Suggestion: First find an expression for the wave speed at any point a distance x from the lower end by considering the rope’s tension as resulting from the weight of the segment below that point.