Understanding the Principle of Superposition for Mechanical Waves

[Craig113]

I would like to understand exactly how to principle of superposition works. I don’t think its so simple that you could imagine as first.

A mechanical wave doesn’t need a physical movement to spread thorough a medium, and it can not be stopped by physically stopping the movement of some of the oscillators that make up the medium it travels through.

Let me clarify what I mean with the help of an example. You start a mechanical transversal wave in a feather by pulling one end of the feather to the side and back again. This will creates a mechanical wave that travels with the same appearance through the feather.. but some part of the feather is physically unmoveable, while the rest of the feather after that is moveable, what will happen? I think that the mechanical wave can pass through the unmoveable part to regain its movement on the other side of the obstacle.

If this is not the case, just don’t get it. Because if mechanical waves only spread through physical movement of the oscillators, then why can't a much bitter mechanical wave simply permanently destroys a smaller when they pass each other with opposite peaks? I mean it should simply “kill” the driving force behind it at the expense of loosing some of its own driving force, but that’s not what happens.

So how does it work anyway? How does the mechanical waves spred without actual movement of the oscillators ?

 

[Crosson]

Good question, I have never thought about it before. I put it in the context of two pulse waves traveling down a string with equal but opposite amplitudes. During the time that they "collide" the entire string is perfectly straight, and then the pulses continue on past each other.

What is the difference between string at the moment of superposition and a string with no wave pulses traveling at all?

I think the answer has to do with the internal forces in the string. Imagine that the pulses are supported by a force vector perpendicular to the string, acting at the center of mass of the wave form. This force rigorously represents the internal forces in the string (because of the center of mass theorem). Now, imagine that it is this force vector which travels down the string, and the pulse waveform that the string takes on as secondary.

In this view, it is easy to see what is happening at the moment of superposition; two equal but opposite forces are acting at the same point and producing a net force of zero. The important thing is that the force vectors don't effect each others movement, they just continue traveling down the string. This is literally what happens, its just that the vectors provide an easy way to visualise the internal forces of the string

I realize this does not address your question on a molecular level, but on the level of continuous matter the answer to your question has to do with the internal forces in the string.

 

[Chi Meson]

It seems you have a misconception here. A mechanical wave propagates ("spreads" as you said) through particles, and these particles DO move as harmonic oscillators. The term "transverse" that you use in regards to your feather example (an odd example, by the way) specifically means that the direction of particle motion is perpendicular to the direction of wave propagation.

If two waves pass through the same material and cause superposition such that they cancel ("destructive interference") many folks will say "where did the energy go?" The problem is that often they forget that the medium, although appearing "flat" at the moment of destructive interference, the medium is still in motion carrying all the kinetic energy with it.

I can't tell if this helps at all. Perhaps you can clarify your question further.



edit: Crosson posted while I was writing.

 

[Antiphon]

When you take a "snapshot" of the string you are observing only it's displacement.
There is a momentum that is non-zero distributed over the string.

And all this sort of superposition can be made to happen mechanically.
A simple example.

You have a string tied to a wall. You shake it and waves move down the string,
bounce off the wall and come back.

But if you tie it to a special spring-and-mass device on the wall that can abosorb
the incoming wave, there will be no reflected wave. Work will have been done on the
device which "abosrbed" the reflected wave.

But it is valid to consider that the device transmitted a wave equal and opposite
back toward you which canceled the reflected wave by superposition.

 

[Craig113]

I still have not recived any acceptable answer to the question.

 

[Antiphon]

A mechanical wave doesn’t need a physical movement to spread thorough a medium, and it can not be stopped by physically stopping the movement of some of the oscillators that make up the medium it travels through.

This is incorrect. A mechanical wave is by definition one part physical
movement (the kinetic energy part) and one part forces (the potential
energy part).

Let me clarify what I mean with the help of an example. You start a mechanical transversal wave in a feather by pulling one end of the feather to the side and back again. This will creates a mechanical wave that travels with the same appearance through the feather.. but some part of the feather is physically unmoveable, while the rest of the feather after that is moveable, what will happen?

Please explain this a bit more clearly. Why is part of the feather unmovable?

I mean it should simply “kill” the driving force behind it at the expense of loosing some of its own driving force, but that’s not what happens.

This is what happens. It is called desructive interference. The wave
amplitude can disappear as long as the energy and momentum are
conserved.

So how does it work anyway? How does the mechanical waves spred without actual movement of the oscillators ?

It requires at least one but usually both of the following two things:
1) Actual movement of the oscillators
2) Non-zero forces on the oscillators.

If you shake a rope a certain way, there could be a point in the center
of the rope which doesn't move. Perhaps this is what you mean with
the feather.

But that part of the rope still transmits forces to the other parts of the
rope which do move. That's how energy moves through a node in a rope.
It's not kinetic energy at the node, its potential energy.