Stationary waves in a vertical rope

In summary, the conversation discusses the possibility of producing stationary waves in a vertical rope and whether the simple wave function equation can still be applied in this scenario. The result obtained for the wave equation and the stationary wave equation are also mentioned, along with the uncertainty about the acceleration and direction of the rope. A potential approach for analyzing the problem using a partial differential equation is also suggested.
  • #1
jaumzaum
434
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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:
$$y_{stationary}=2A sin(w\sqrt{4x/g}+\phi) cos(wt)$$
 
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  • #2
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.
 
  • #3
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##.
 
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Related to Stationary waves in a vertical rope

1. What are stationary waves in a vertical rope?

Stationary waves in a vertical rope are a type of standing wave that forms when two waves with the same frequency and amplitude travel in opposite directions along a fixed medium, such as a rope. This results in a pattern of nodes (points of no displacement) and antinodes (points of maximum displacement) that do not move along the rope.

2. How do stationary waves in a vertical rope form?

Stationary waves in a vertical rope form when two waves with the same frequency and amplitude travel in opposite directions along a fixed medium, such as a rope. As the waves interact, they interfere with each other and create a pattern of nodes and antinodes that do not move along the rope.

3. What factors affect the formation of stationary waves in a vertical rope?

The formation of stationary waves in a vertical rope is affected by several factors, including the frequency and amplitude of the waves, the tension and density of the rope, and the length of the rope. These factors determine the wavelength and speed of the waves, which in turn affect the pattern of nodes and antinodes that form.

4. What is the significance of stationary waves in a vertical rope?

Stationary waves in a vertical rope have many practical applications, such as in musical instruments like guitars and violins, where they produce distinct harmonics. They also have important implications in fields such as acoustics, optics, and quantum mechanics, where they can be used to study wave behavior and phenomena.

5. How are stationary waves in a vertical rope different from traveling waves?

Stationary waves in a vertical rope are different from traveling waves in that they do not propagate or transfer energy along the medium. Instead, they remain in a fixed position and oscillate in place. Traveling waves, on the other hand, move through the medium and transport energy from one point to another.

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