Nodes in a standing wave and energy transmittance

[itchybrain]

I am sure this has been answered many times. But I've read about 40 posts on standing waves, and I still have a "standing" question.

I am having a hard time visualizing how energy can be transmitted at a node of a standing wave. Basically, how can an immobile point be pulling on its neighboring points, without being in motion itself?

Let's take a simple case. Imagine a "rope" composed of single atoms, where one atom is connected on the left to exactly one atom, and to exactly to another atom on the right.

When the middle atom is pushed down, since it is has bonds to the neigboring atoms, it drags these neighbors down. This is visually manifested as a wave. For simplicity's sake, let's follow this wave in one direction (to the right only, for example). This middle atoms pulls on neighbor # 1, which pulls on neighbor # 2, and so on. The wave propagates. The original middle atom eventually loses kinetic energy, or is pulled back towards equilibrium/center, and returns to it's original spot.

But this whole phenomenon relies on the middle atom moving down and tugging on it's neighbor.

Now we go to the nodes on a standing wave. At this point, there is no movement, as this point is simultaneously exposed to forces that equally drive it up and down, therefore cancelling. So this point at the node does not move. And yet, it's neighbors will get pulled: for example, the neighbor on the left will get pulled up, and the neighbor on the right will get pulled down, SIMULTANEOUSLY!

I am sure I am missing a fundamental concept here, but it is very counterintuitive to me. For example, I get how in Newton's third law, the action-reaction force pairs do not cancel because they are acting on different objects. But transmittance at the node continues to elude me.

Can anyone explain this to me (or point me to a high-yield post where this is explained in simple terms)?

 

[Dale]

I am having a hard time visualizing how energy can be transmitted at a node of a standing wave.

Energy isn't transmitted at a node of a standing wave. That is why it is a standing wave, the energy stays "standing" in place.

Here is a little write-up on the topic:
http://physics.usask.ca/~hirose/ep225/lecture-17.pdf

 

[itchybrain]

It seems like this has been asked before in the past, but still no satisfactory answer. For example, see the past thread:
How do nodes on a string produce tension if they are stationary?

Your write-up on the topic mentions that energy is confined between nodes. Is it fair to say that nodes are acting like the fixed point where a rope is is tied to a wall, and a wave traveling along this rope bounces back and reflects (but inverted)?

 

[tech99]

A standing wave is not transmitting energy along the rope. But if there is some energy loss in the system, and it is excited from one end, the node becomes imperfect and we see small movement taking place. We see the same thing with electrical waves on a wire - if there are losses, there is a small current at a node rather than zero..This is because there is now a traveling wave.

 

[sophiecentaur]

Real standing waves do not have perfect nodes. If the node were 'perfect' then you would have no loss and the wave would have grown to infinite amplitude. But the energy source must have a finite Impedance / Admittance so there must be some damping in the system and the peaks of the standing wave on a transmission line will only be twice the amplitude of the source. Most elementary treatments of the problem will ignore these considerations - causing confusion.

 

[tech99]

It would seem to be possible to supply the losses by applying a small voltage at a node. In this case the peaks of the standing wave will be many times greater than the voltage of the generator.

 

[Delta2]

In a perfect standing wave there is no force neither energy transmition from a point to its adjacent points. There is energy transmission only during the initial phase where the standing wave is formed , starting from a string at a straight line in which we have to supply some energy in order to set its points in a motion pattern of the standing wave. After the initial phase the points continue to move doing independent oscillations without exchanging energy or forces.

 

[Delta2]

Well maybe i was partially wrong, better take a carefull look at this derivation and figure http://en.wikipedia.org/wiki/Vibrating_string#Derivation. You see in the figure that each infinitesimal segment of the string(so and an infintesimal segment centered at a node) has two forces acting on it, from left T1 and from right T2. Since the infinitesimal segment at a node isn't moving at all it will be T1+T2=0 at all times, however this doesn't necessarily implies that T1=T2=0. In general it will be T1,T2≠0. And so by Newtons third law, the node exerts force -T1≠0 on the left and force -T2≠0 on the right, though the node isn't moving.

 

[sophiecentaur]

It would seem to be possible to supply the losses by applying a small voltage at a node. In this case the peaks of the standing wave will be many times greater than the voltage of the generator.

I'm not sure what you have in mind here but, with a transmission line, fed from a source of power, injecting an extra signal into a region around a node would merely involve altering the total power injected. The size of the standing wave would then depend upon the source impedance of the extra signal and it could be more or less (i think) depending on the phase relationship between the two sources. In the normal case, power is dissipated in the source impedance and in the suggested case, power may be dissipated in within both sources. It is not a simple situation.