Serway & Jewett — Wave Motion: Problem 48

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 \(\Delta t \approx 2\sqrt{L/g}\). 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.