# Waves - I just need to confirm an idea

Hello everyone!

I've got a question from the book "Fundamentals of Physics/Halliday, Resnick, Walker" - 6th ed, page 395, #23P.

A uniform rope of mass m and lenght L hangs from a ceiling.

(a) Show that the speed of a transverse wave on the rope is a function of y, the distance from the lower end, and is given by v = sqr(gy).

(b) Show that the time a transverse wave takes to travel the length of the rope is given by T = 2sqr(L/g)

***

(a) Follows from the wave speed on a string definition - v = sqr([tau]/[mu]), where [tau] = mg, and [mu] = m/y --- I think that's right.

(b) I think that the solution might follow from the period of a simple pendulum --- T = sqr(L/g)... but I'm not quite sure how the 2 comes up... it's just a guess.

I hope you guys can give a hand.

Thanks a lot.

## Answers and Replies

Ambitwistor
For (b), you have v = dy/dt = &radic;(gy),

T = integral dt = integral dy/&radic;(gy)

where the latter integral is taken as y ranges from 0 to L.