Graduate How Can the 1D Wave Equation Describe the Dynamics of a Rope Snap?

[Winzer]

I was wondering:
Suppose I have a rope on a table with one end fixed. I take the other end and give a quick and hard snap. It will produce a traveling wave. Is there a PDE that could describe this? Or is there one that I could come up with? The time evolving amplitude will probably depend on the material of the rope(some constant), and the initial force of the snap.

 

[Marin]

How about the 1D wave equation?

\frac{\partial^2\Psi}{\partial z^2}=\frac{1}{v^2}\frac{\partial^2\Psi}{\partial t^2}

where

\Psi(z,t)=Aexp(\omega t-kz)

A - amplitude
omega - circular frequency
k - wave number
x, t respectively space and time coordinates
v - the propagation velocity

Solving the equation uses Fourier Series. This is where you have to use your fixed end (the Series will consist only of sine terms). The "snap" will give additional term(s) causing inhomogeneity.

The rope material and its density function determine the propagation velocity (cf. waves throuth elastic mediums). I think the amplitude has nothing to do with them.

all the best,

marin