Stationary waves in a vertical rope

[jaumzaum]

I was wondering if we could produce stationary waves in a vertical rope. There is a nice result we can get from a vertical rope that the pulse created from the lower extremity travels upwards with acceleration g/2 and the pulse created in the upper extremity travels downwards with acceleration -g/2. I was trying to get the equations of the stationary waves in a vertical rope (that is, if they exists), but I don't know if the most simple wave functions hold anymore.

For example, I was taught any wave can be written in the form:
$$y=f(x-vt)$$
But here the velocity is variable, so can we still write the above equation?
Also, will the amplitude remain constant?

I will write the result I got, but I don't know if they are correct, can anyone help me figure this out?
For the wave:
$$x=\frac{gt^2}{4}$$
$$\alpha=w\sqrt{4x/g}-wt+\phi$$
$$y=A sin(w\sqrt{4x/g}-wt+\phi)$$

For the stationary wave:
$$y_{stationary}=2A sin(w\sqrt{4x/g}+\phi) cos(wt)$$

 

[anuttarasammyak]

I am not sure of acceleration you state. Wave propagates with increasing speed ?
I am not sure vertical, horizontal or any direction of rope length make difference of the vibration.

 

[etotheipi]

I haven't checked this but maybe you can analyse it with something like$$\frac{d}{dx} \left(T(x) \frac{\partial y}{\partial x} \right) = \mu \frac{\partial^2 y}{\partial t^2}$$where I have taken $\hat{x}$ pointing down and $\hat{y}$ pointing to the right. We could approximate that $T(x) \approx \mu g(L-x)$, and try and solve the PDE w/ boundary conditions $y(0) = y(L) = 0$.