Why Does the String Behind the Node Stop Oscillating in a Standing Wave?

[Eric_meyers]

Homework Statement

"A standing wave in the form of a string attached to a driven tuning fork is created. We then move the furthest boundary point to a node somewhere along the string. The node of course was originally at rest by definition. The string behind the node with the boundary point ceases to oscillate. Why?

Homework Equations

The Attempt at a Solution

So in this problem I was thinking of Newton's law of equal and yet opposite force, but the node was already at rest so with no motion I couldn't utilize this concept. I'm drawn between using some concept of energy, that the boundary point is creating a point of discontinuity in the medium so the wave can't propagate beyond it... but I'm not sure exactly how it creates this point of discontinuity.

Is it absorbing the energy? How? I'm drawing a big blank because I don't see the boundary point doing work on the string since the net displacement is 0.

 

[turin]

I'm pretty sure that the boundary point does no work on the string; as you say.

This is a good question. I don't know what the answer is "supposed to be", but I'm guessing that they want you to decompose the standing wave into traveling waves, and then consider the effect of the boundary condition on the traveling waves.

 

[tomcue]

I would approach this problem by first understanding the basics of standing waves. A standing wave is formed when two waves with the same frequency and amplitude travel in opposite directions and interfere with each other. This results in a stationary pattern of nodes (points of no displacement) and antinodes (points of maximum displacement).

In the given scenario, the tuning fork is driving the string to oscillate at a certain frequency. The boundary point, which is now moved to a node, is essentially creating a fixed end for the wave. This means that the wave cannot propagate beyond this point, as the boundary point is not allowing any further displacement of the string.

To understand why the string behind the node with the boundary point ceases to oscillate, we need to consider the energy of the system. The tuning fork is continuously supplying energy to the string, causing it to vibrate. However, when the boundary point is moved to a node, it creates a point of reflection for the wave. This means that the energy of the wave is being reflected back towards the tuning fork instead of being transmitted through the string. As a result, the string behind the node experiences a decrease in energy and eventually stops oscillating.

In conclusion, the boundary point creates a point of reflection for the standing wave, causing a decrease in energy and resulting in the string behind the node to cease oscillations. This concept can also be explained using the principle of superposition, where the reflected wave from the boundary point interferes with the original wave, resulting in a cancellation of energy and a stationary wave pattern.