How do string waves pass through each other without interfereing?

[learning]

From what I have learned so far a wave propogates on a string by having the point on the string (that is closest to the wave and in the direction the wave is moving) move upwards or downwards depending on the wave and this point moving due to tension causes the next to move simillarly and thus the wave propogates through the string. My problem comes from when there is let's say an equal yet oppisite wave moving on the string and they meet. According to me at this point because the point in question will have equal reason to move both up and down it will stay in position and the both waves will collapse or be reflected I don't know. I know that I am wrong as the waves do pass through each other I just want to know why? thanks for your help.


PS: This is my first post though google has lead me here for years.

 

[olivermsun]

The equation for the wave on a string is "linear" for waves of small amplitude. This means that the sum of two solutions to the equation is also a solution of the equation.

Physically this means that you can have two (or more) waves on the string, and you can treat the waves completely separately—the displacement, the velocities, forces, everything—the waves have no interaction with one another. When you add them together, all the physics of the string are still satisfied.

 

[rtsswmdktbmhw]

My problem comes from when there is let's say an equal yet oppisite wave moving on the string and they meet. According to me at this point because the point in question will have equal reason to move both up and down it will stay in position and the both waves will collapse or be reflected I don't know. I know that I am wrong as the waves do pass through each other I just want to know why?

The wave is not reflected in your scenario - reflection occurs when the wave reaches the end of the rope, bounces back, and starts coming towards you.

You are right that the waves can cancel each other out so that you'll end up with no displacement in the rope. See diagram for destructive inteference: http://en.wikipedia.org/wiki/Interference_(wave_propagation)#Mechanism

But, just because the rope is not displaced doesn't mean that the wave has disappeared. The energy of each wave continues traveling along the rope.

 

[CWatters]

Perhaps one way to think of it...

When equal but opposite waves meet the net force on that point is zero so there is no displacement but the local tension in the rope increases.

 

[Philip Wood]

Suppose two equal and opposite gaussian pulses moving in opposite directions have their centres at the same point P on the rope at one instant. The displacement of the rope will be zero at the region of overlap (and everywhere else). But the (transverse) velocity of the rope will not be zero in the region of overlap. On either side of P the transverse velocities of the rope will be in opposite directions. You can convince yourself of this by drawing diagrams and thinking about which way the rope is moving as the pulses go though. Thus the waves can continue to propagate and will pass through each other unscathed, as the Principle of Superposition predicts.

 

[learning]

Thanks for your answer. but i still do not understand how the wave can effect the next point in the string if the point on the wave closest to said next point in the string has a zero acceleration,velocity and displacement(as both waves cancel each other out.) I guess its another one of those things that I just have to accept as true and move on.

 

[olivermsun]

The acceleration, velocity, and displacement of a wave are not all in phase with each other. They have a definite phase relationship (acceleration -> velocity -> displacement ~ acceleration).

The two waves are traveling in opposite directions, so the acceleration, velocity, and displacement have the opposite phase relationships and wouldn't cancel out at all the same points, at the same time.

 

[CWatters]

Thanks for your answer. but i still do not understand how the wave can effect the next point in the string if the point on the wave closest to said next point in the string has a zero acceleration,velocity and displacement(as both waves cancel each other out.) I guess its another one of those things that I just have to accept as true and move on.

Read the reply by Philip wood again.

 

[rtsswmdktbmhw]

Thanks for your answer. but i still do not understand how the wave can effect the next point in the string if the point on the wave closest to said next point in the string has a zero acceleration,velocity and displacement(as both waves cancel each other out.) I guess its another one of those things that I just have to accept as true and move on.

Think about it this way. You and a friend are pushing against each other with equal force (as hard as both of you can). So overall, neither of you are moving - acceleration, velocity and displacement are all zero.

Now both of you suddenly slide to the right (i.e. in opposite directions). Each of you still pushes as hard as you can. What happens? Will you fall over/lose your balance, or do you think you will simply remain upright without moving?

(You and your friend are analogous to particles in a rope.)

 

[mal4mac]

Think about it this way. You and a friend are pushing against each other with equal force (as hard as both of you can). So overall, neither of you are moving - acceleration, velocity and displacement are all zero.

Now both of you suddenly slide to the right (i.e. in opposite directions). Each of you still pushes as hard as you can. What happens? Will you fall over/lose your balance, or do you think you will simply remain upright without moving?

(You and your friend are analogous to particles in a rope.)

Nice example, but it doesn't quite get us there. A rope has a multitude of particles all interacting with each other. It's far from obvious that all those particle interactions will give the result we observe. If the textbook you are using just asks you to accept this I think just "accepting this and moving on" is a good idea - while keeping rtsswmdktbmhw's example in mind as a motivating example, and also keeping in mind that if you ran a simulation taking all particles into account you'd get the result observed. (You may doubt this - OK take a few months and develop the simulation, develop nice visuals, and stick it on the web! Nice resource for students...)

 

[DabblingMathmo]

Perhaps one way to think of it...

When equal but opposite waves meet the net force on that point is zero so there is no displacement but the local tension in the rope increases.

I just wanted to pick up on and query this, specifically the local tension increases...

If we had two equal but opposite propagating pulses on an elastic string that encounter each other, then at the instant where they completely cancel any and all transverse displacement then I don't see how can there be any enhancement of the local tension?

When you formulate the (linear) equations the tension force goes as the second derivative of the transverse displacement with respect to the string's equilibrium axis which would be zero. Furthermore on an intuitive level this seems wrong to me, if there is no deformation of the string then how can there be any sort of enhancement of tension at that instant?

After the temporary destructive interference, then the continued propagation of the pulse is more so to do with the waves energetics (tying in with Philip Woods discussion) than because of any localised tension.

Anyway, I am fairly sure of this so just wanted to (a) clarify that it is not tension for the OP or (b) have someone explain to me why I am wrong because if so I have some fundamental misunderstanding to iron-out. Cheers,

 

[KL7AJ]

This is a beautiful question...and I love demonstrating this in my radio classes. What you observe is known as the Superposition principle, which is utterly simple in terms, but has tremendous implications. The superposition principle says this. At any point in space there is one and only one amplitude possible. (This applies to electrical potential, mechanical position, or any number of physical phenomena). This also is what allows a loudspeaker to produce a nearly infinite number of audio frequencies simultaneously.

Think how strange the universe would be if superposition did not hold!

One caveat...the system must be LINEAR for Superposition to strictly apply...but in most of nature we see superposition work.

Since it's a fundamental principle, it's really hard to define in terms of anything else...it is almost a definition by itself...and we know better than to argue with a definition!

Hope this gives some insight.

Eric

 

[olivermsun]

After the temporary destructive interference, then the continued propagation of the pulse is more so to do with the waves energetics (tying in with Philip Woods discussion) than because of any localised tension.

Anyway, I am fairly sure of this so just wanted to (a) clarify that it is not tension for the OP or (b) have someone explain to me why I am wrong because if so I have some fundamental misunderstanding to iron-out.

Well you are right. As Phillip Wood explains, consider two equal and opposite Gaussian pulses moving in opposite directions. At the moment when the pulses are perfectly superposed, the displacements will cancel completely and the string will be flat. But the transverse velocities are not zero even at the moment of cancellation.

Suppose for example you have a "positive" pulse traveling from left to right and a "negative" pulse traveling from right to left. Just before the moment of cancellation, the displacement left of center is positive going to zero and the displacement right of center is negative going to zero. Right after the moment of cancellation, the section to the left is continuing from zero to negative displacement and the section to the right is continuing from zero to positive. What this implies is that, even at the moment of cancellation, the string to the left has a negative transverse velocity and the string on the right has a positive transverse velocity.

So at the moment that the string is completely flat, the energy of the two waves is carried entirely in kinetic energy (just as with any oscillator crossing its equilibrium position).

 

[KL7AJ]

I hope you get a chance to see my Lecher Wire demo:


Superposition in the "flesh". :)

 

[mal4mac]

If we had two equal but opposite propagating pulses on an elastic string that encounter each other, then at the instant where they completely cancel any and all transverse displacement then I don't see how can there be any enhancement of the local tension?

Think of the two kids pushing each other - all movement is canceled but there is a lot of tension there (perhaps in more way than one .

 

[DabblingMathmo]

Think of the two kids pushing each other - all movement is canceled but there is a lot of tension there (perhaps in more way than one .

Thanks for the reply mal4mac, but I don't think there is actually any tension the cartoon you describe, there might be a system of applied forces which sum to zero, but no tension arises (think of tension as an elastic property?).

Anyway, I think the key thing here is OP is asking why the waves don't collapse during the interaction, and why it is that they pass through each other? This has been answered by other posters (only the displacement cancells out, other properties are out of phase). I just wanted to clarify that if there is no transverse displacement, there is no tension perturbation/enhancement, so it isn't the relevant quantity that keeps the waves in motion after the interaction.

 

[Philip Wood]

The standard first order theory of transverse waves on a stretched spring (that is the treatment that predicts linearity, and a wave speed independent of frequency, or, equivalently, of pulse shape) ASSUMES constant string tension. The propagation mechanism is by force components transverse to the string. These arise from the string not being straight. See thumbnail.

In the case we're considering, of equal and opposite pulses occupying, at one instant, the same portion of string, the string is instantaneously straight, but, as I tried to explain earlier, the string has transverse velocities. Another case where the string is instantaneously straight, but moving is when it has just been hit by a hammer. We know that even if the hammer is immediately withdrawn, the string will subsequently deform, and waves will propagate away from the point where the hammer hit the string.

PS Liked the Lecher wires.

 

[learning]

Than you guys for your replies. I think I understand the situation a bit better now.

 

[mal4mac]

Thanks for the reply mal4mac, but I don't think there is actually any tension the cartoon you describe, there might be a system of applied forces which sum to zero, but no tension arises (think of tension as an elastic property?).

Yes, to be exact it is *compression*. For tension, think of the kids pulling each other rather than pushing each other. Tension doesn't only apply to elastic bands. Take a look at the wikipedia page:

https://en.wikipedia.org/wiki/Tension_(physics)

The picture at the top right shows tension in a rope during a tug of war - and the rope isn't elastic (Could be more fun if it was .