What Power is Needed for Sinusoidal Waves in a Taut Rope?

[MarkFL]

Hello, MHB Community! (Wave)

anemone has asked me to stand in for her for a few weeks, so please be gentle. (Bigsmile)

Here is this week's POTW:

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A taut rope has a mass $M$ and length $L$. What power must be applied to the rope in order to generate sinusoidal waves having an amplitude $A$ and wavelength $\lambda$ and traveling with speed $v$?

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Remember to read the https://mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to https://mathhelpboards.com/forms.php?do=form&fid=2!

 

[MarkFL]

No one answered this week's problem, and my solution is as follows:

Spoiler

A formula for power $P$ we can apply here is:

\(\displaystyle P=\frac{1}{2}\mu\omega^2A^2v\)

Where:

\(\displaystyle \mu=\frac{M}{L}\) and \(\displaystyle \omega=\frac{2\pi v}{\lambda}\)

Hence:

\(\displaystyle P=\frac{1}{2}\left(\frac{M}{L}\right)\left(\frac{2\pi v}{\lambda}\right)^2A^2v=\frac{2\pi^2A^2Mv^3}{L\lambda^2}\)