# Transverse Wave on a Hanging Cord

[SOLVED] Transverse Wave on a Hanging Cord

Problem. A uniform cord of length L and mass m is hung vertically from a support. (a) Show that the speed of tranverse waves in this cord is $\sqrt{gh}$ where h is the height above the lower end. (b) How long does it take for a pulse to travel upward from one end to the other?

For (a), I know that the speed of a transverse wave on a cord is given by $v = \sqrt{T/\mu}$ where T is the tension on the cord and $\mu$ is the linear density. As far as I understand, T = mg and $\mu = m/L$ so $v = \sqrt{gL}$. Now, unless h = L (which I know isn't), I don't see how h plays a role here.

The answer to (b) is just L/v obviously.

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Doc Al
Mentor
Realize that the tension varies along the length of the cord.

Let's measure the cord from the bottom up with the bottom being y = 0 until y = L. Then tension on the cord at y is $\mu y g$. Is that what you mean when you wrote that the tension varies along the cord?

Doc Al
Mentor
Absolutely. Using the terminology of the problem statement, $T = \mu g h$.

Ah, OK. I get it now. I thought h was some constant like L, not a variable.