Why is a mechanical wave inverted at a boundary?

[Freddy86]

Hi, please could someone help clarify the reason why a mechanical wave is inverted at a boundary as I'm really stuck! Some sources I have read seem to suggest it can be explained by Newton's 3rd law whilst others suggest its to do with conservation of momentum.

Newton's 3rd law - consider the crest of a wave pulse of a rope approaching a fixed end. The last particle of the rope will receive an upward displacement which exerts an upward force on the first particle of the fixed boundary. This is an equal and opposite reaction and the wave is inverted and reflected. Is that correct?

Conservation of energy - the rope supposedly has a forward momentum. When the rope hits the boundary in order to conserve momentum an inverted wave is reflected back. I don't understand this as how can a rope wave have mass (and therefore momentum) as the motion is only the propagation of energy (the particles themselves move perpendicular). Can energy not be conserved by the fixed end (+ earth) receiving momentum as in the case of a tennis ball hitting the wall. I don't see the need for the wave to be inverted to conserve energy?

I know its a bit long but I would greatly appreciate some help.

 

[Meir Achuz]

The rope is fixed at the boundary. This means the reflected wave must cancel the incident wave, and thus be its negative.

 

[rexregisanimi]

I hope Meir's answer was sufficient but let me give my own words (if it helps at all).

At a fixed boundary, the rope must remain fixed in place (no vertical motion). So as the "incoming" wave hits the boundary, another wave (the "outgoing" wave) must be formed in such a way as to cancel the "incoming" wave (that way there is no net displacement). This "outgoing" wave (the reflection of the first wave) is thus the opposite of the "incoming" wave (the top of the incoming wave will now be at the bottom of the outgoing wave and the left of the outgoing wave will be what was the right side of the incoming wave).

This YouTube video is kind of neat in a related sort of way:



...and a video of actual strings:

 

[sophiecentaur]

If, instead of a perfect clamp, you imagine a damping paddle that will absorb all the energy by being damped in oil, say (a perfect termination) and the paddle will have exactly the same displacement as the incoming wave. There is no energy left to be reflected. Now imagine an imperfect termination, which will not absorb all the incident energy. Some has to be reflected. The amplitude of the displacement of the paddle will now be a bit less. It will be equal to the amplitude of the incoming wave minus the amplitude of the reflected wave (there's your inversion). When the paddle is replaced by a perfect clamp, the reflected wave will be of equal amplitude and still in anti phase (incident - reflected amplitudes = 0)

 

[PJC01]

The reason why a mechanical wave is inverted at a boundary can be explained by both Newton's 3rd law and conservation of energy.

According to Newton's 3rd law, for every action, there is an equal and opposite reaction. In the case of a wave approaching a boundary, the particles at the boundary will experience a force from the particles in the wave. This force will be in the opposite direction of the wave's motion, causing the wave to be inverted and reflected back. This can be seen in the example of a rope wave approaching a fixed end, as mentioned in your question.

In addition, conservation of energy also plays a role in the inversion of a mechanical wave at a boundary. When a wave reaches a boundary, it transfers its energy to the boundary. In order to conserve energy, the reflected wave must have the same amount of energy as the original wave. However, the reflected wave is moving in the opposite direction, so it must have an inverted shape in order to maintain the same amount of energy.

As for your question about the momentum of a rope wave, it is true that the particles in the wave do not have mass and therefore do not have momentum. However, the wave itself does have momentum as it is carrying energy. This momentum is transferred to the boundary upon reflection, again following the principle of conservation of momentum.

In summary, the inversion of a mechanical wave at a boundary can be explained by both Newton's 3rd law and conservation of energy. Both principles work together to ensure that energy and momentum are conserved in the reflection of the wave.