How can the wave speed of a rope with negligible stiffness be determined?

[Corneo]

Hi I am working on a certain homework problem and I would appreciate some hint or inputs.

A rope, of length $L$, is attached to the ceiling and struck from the bottom at $t=0$. The rope has negible stiffness, how long would it take for the wave to travel up the string and back down?

I have worked on the problem for a while and concluded that the velocity will vary because tension in the rope varies as you travel along the medium.

Tension, $T$, can be written as the distance from the bottom of the rope. That is $T(x)=x \mu g$, $x$ is the distance measured from the bottom of the rope; and $\mu=m/L$ is the linear mass density.

This is where I am lost. I think the wave equation is where I should start off, but not sure how to apply it to this problem

$$\frac {\partial ^2 y }{\partial t^2} = v^2 \frac {\partial ^2 y}{\partial x^2} = \frac {T(x)}{\mu} \frac {\partial ^2 y}{\partial x^2}= \frac {x \mu g}{\mu} \frac {\partial ^2 y}{\partial x^2} = x g \frac {\partial ^2 y}{\partial x^2}$$

Any hints or inputs would be appreciated.

 

[Clausius2]

Do you really think the gravitational force plays an important role in this problem?. You mentioned the rope is "struck". How about the magnitude of the strike?. I'm lost because you don't mention about it again. If you have the figures, compare the magnitude of the tension caused by the strike and the tension you've just calculated above.

 

[Corneo]

I do think that gravitational force plays a part in this problem since it attributes to the tension in the rope. At the bottom of the rope there is no tension and at the top of the rope the tension is the greatest.

 

[Clausius2]

Do you have figures to compare?. For example, How long is the rope?

Moreover, What is the value of the force or the external interaction applied at the bottom as you said :

...and struck from the bottom at ...

 

[Corneo]

No there are no figures. I paraphrase the question and it actually reads "struck sharply from the bottom". The length is still L.

 

[Clausius2]

What is causing the wave propagation?. The simple action of gravity cannot cause the wave propagation. There have to be another external force at the bottom. Post your boundary conditions.