Understanding Standing Waves and their Velocity on a String

[jlyu002@ucr.e]

Is a standing wave actually a traveling wave appearing to be still do to interference of two different waves?

If the above statement is true, how can a resultant wave even form? The wave traveling forward is not in or at the same time as the wave reflecting back. (How can there interference when the wave traveling back is not lined up with the wave traveling forward)

Perhaps they just appear to interfere.

Lastly, is the velocity for a certain string the same for a standing wave.

 

[Simon Bridge]

Is a standing wave actually a traveling wave appearing to be still do to interference of two different waves?

A standing wave is a result of interference - it does not "appear" to be still, it is still (in the sense that it has no translational motion).

If the above statement is true, how can a resultant wave even form? The wave traveling forward is not in or at the same time as the wave reflecting back. (How can there interference when the wave traveling back is not lined up with the wave traveling forward)

It happens when the interfering waves are exactly the same speed and wavelength - the intereference cancels out the translational motion. You can see the effect of the motion of the two traveling waves in the way the standing wave oscillates.

If you like you can construct the standing wave from the interference of two traveling waves by hand in a series of sketches. It's actually easiest to do with a triangle-shape wave on a length of string fixed at both ends and displaced in the middle. That wave should be decomposed into two equal waves that add to the initial displacement, headed in opposite directions. Take time steps so that the wave travels 1/4 the string length. Remember to invert on reflection.

Perhaps they just appear to interfere.

No - any two waves occupying the same space will interfere. That is what interference is. It is not some kind of illusion.

Lastly, is the velocity for a certain string the same for a standing wave.

If, by "velocity for a certain string" you mean the wavespeed for that string, then no - the wave-speed of a standing wave is zero.

You can work out the dynamics of a standing wave from two traveling waves algebraically to see the relationships. There are also lots of demonstrations online.

 

[jlyu002@ucr.e]

Hi Simon, what if we were to look at a standing wave, and only look at the top part of the wave, the non inverted wave. Could we say that that part has a velocity? And can we say that the net speed of the string is zero?

I'm saying this because in my book there is a problem that states, " The second-harmonic wavelength for a rope fixed at both ends is .5m. How fast do transverse waves travel along this rope if the fundamental frequency is 4Hz?

When I watched some youtube videos for standing wave, I noticed that the wave first propagates and hits the wall and then the inverted wave takes over. Based on that illustration, how can there be two superimposed opposite waves at the same time, creating interference?

Perhaps, due to this action happening so fast, due to the specific wavelength and frequency, and due to the phenomena where the two waves are inverse of each other, we see one motion in which the string is just bouncing up and down at the anti nodes.

Sorry Simon, I thought I only had one question, but thinking about this and your statements above prompted me to write my thoughts out.

Thank you Simon for all your help. You are seriously one of my physics heros!

 

[Simon Bridge]

Hi Simon, what if we were to look at a standing wave, and only look at the top part of the wave, the non inverted wave. Could we say that that part has a velocity? And can we say that the net speed of the string is zero?

Yes. The string only moves up and down - the translational (along the string) velocity is zero.
It is unclear what you mean by "the top part" though.
If you are referring to the speed of the components, then those are traveling waves, not standing waves, so they have a non-zero speed.

I'm saying this because in my book there is a problem that states, " The second-harmonic wavelength for a rope fixed at both ends is .5m. How fast do transverse waves travel along this rope if the fundamental frequency is 4Hz?

That question is asking about the wave speed for traveling waves.

When I watched some youtube videos for standing wave, I noticed that the wave first propagates and hits the wall and then the inverted wave takes over. Based on that illustration, how can there be two superimposed opposite waves at the same time, creating interference?

Please provide a link s I know what you are talking about. It sounds to me like the video showed a pulse which was much shorter than the length of the string. To get a standing wave, you need the wavelength to be bigger than the length of the string or two wave generators.

Thank you Simon for all your help. You are seriously one of my physics heros!

So - no pressure huh?

 

[Simon Bridge]

Just to be sure we are talking about the same thing: I have an animation for you from this page:
http://www.physicsclassroom.com/mmedia/waves/swf.cfm

The blue and green waves are the interfering traveling waves being generated in some manner (either reflecting off a boundary or using two wave generators). The black line is what you actually see - that's the standing wave.

Notice how the black line does not travel left or right, though each part goes up and down?
Notice how the black wave has max amplitude when the traveling waves overlap (constructive interference) and a minimum when the traveling waves are inverses of each other (destructive interference)?

 

[jlyu002@ucr.e]

OOO wow this is amazing! Thank you Simon! This was exactly the picture I needed to see! YESSS! Today is a great day!

 

[Simon Bridge]

In the animation, the ends of the string (or whatever) are not in the picture.
Typically it's much faster than that so the black line blurs out.


Notice that the motion does not have to be a standing wave - it can be quite complicated?
... this demo includes slow motion so you can compare with the ideal case in the animation.

BUt here's a nice demonstration combining your older observations with your newer understanding:

... watch the wave approach from the wave generator at the far end of the tank, then you'll see standing waves take over as the inverted wave travels back from the reflector.

A lot of people talk about plucking a string to produce a note. You'll have noticed that the initial shape of a plucked string is not a sine wave - yet all the analysis is done with sine waves. So watch these:



But there's nothing beats actually doing it yourself. Many students don't get hands-on before their 1st year at college - which is silly since the equipment is easy to make, this should be done in secondary school.

 

[jlyu002@ucr.e]

Thanks Simon! These examples definitely reinforced my understanding of the standing wave. One day, I will be as wise as you and get to help others!

 

[davenn]

a great attitude to have
keep learning ... its a lifetime of funDave

 

[Simon Bridge]

...though - don't wait for wisdom to start helping others. Wisdom comes from helping others.

 

[jlyu002@ucr.e]

Sooo wise! I will start! If I can't explain to my grandmother, then I really don't have a great understanding. Talk to you soon Simin as I foreshadow more questions to come.

 

[davenn]

...though - don't wait for wisdom to start helping others. Wisdom comes from helping others.

and the sudden realisation of how little I know haha

 

[Simon Bridge]

I think the "grandmother test" is a bit tough though ;)

Start here - help people with their homework. The neat part with that is you don't need to know how to do it all yourself: just use your experience having done homework before to ask helpful questions. You'll end up working thought he problems with the other person. You have this huge advantage in that you are closer to having to have learned that stuff yourself - it'll be easier for you to understand why they find it difficult, and you'll remember what it was like getting past that.

If you get stuck helping someone you can always ping one of us for guidance ...

 

[jlyu002@ucr.e]

Thanks for the advice Simon. I will definitely keep this in mind!

 

[Philip Wood]

To adapt Wesley's chiasm: Teach Physics until you understand it, then because you understand it, you will teach Physics.

 

[enorbet]

I'm rather fond of the statement that Simon made that "This is what interference is. It is not some form of illusion" since it is so common these days to think of Science as somehow being removed from reality instead of the study of it. In this case not only is interference a measurable quantity by arcane instrumentation, but also by our own ears.

Speaker cabinets are often designed to reduce standing waves which distorts frequency response, generally reducing bass response, but this is rather internal and hidden from casual observers even though they can hear the difference.

My favorite example involves orchestras. Some groups of researchers have experimented with instrument orientations because there are standard locations and they have changed over the years. It was found that many pieces from a given era sounded best when the orientations of the time were used, better than when modern orientations were used. It sounded as if there were more instruments than were present when the appropriate orientation is used. This effect was recorded and analyzed with dynamic instrumentation and sure enough, the orientation caused interference patterns to sound as if another instrument was playing. What the ear heard, and presumably what the original composer empirically understood, was verified by modern recording instruments such as FFTs.

Some of these patterns are subtle and require a trained ear, but others are profound and anyone can hear them. Instrumentation settled any arguments. Often, new instruments have been invented to try to prove/disprove what interferences some have claimed to hear, such as with TIM. Most often, this has resulted in new and better instruments since a trained combination of brain and ear is an exceptionally fine instrument. That is to be expected, perhaps, since no scientific instruments have the processing power of the human brain. Now, if we could only record that.