Mr.p: Good morning. Before we can learn about standing wave patterns,
we first need to understand what happens when a wave encounters an end.
[intro]
In this example the wave pulse is encountering a fixed end;
an end which is fixed in place and will not move.
You can see the wave pulse is reflected and inverted.
Reflected means it changes directions and comes back the other way.
Inverted means the wave pulse is flipped upside down.
...The other option is called a free end.
A free end is free to move up and down as you can see in the video.
When a wave pulse encounters a free end,
the wave pulse is still reflected; however, it is not inverted.
Billy: So, a fixed end causes reflection with inversion,
and a free end causes reflection without inversion. Got it.
Mr.p: Just so you know, in order to illustrate a free end I tied a very long string to the end of the spring.
Okay. Today's demonstrations will involve a string fixed at both ends.
The left end is attached to an oscillator which will be oscillating up and down
in simple harmonic motion at a known frequency.
And the right end is at a constant tension which is caused by a constant hanging mass.
...Okay, let's turn it on and see what happens.
That is a standing wave pattern.
Billy: Cool.
Bobby: Yea?
Bo: I don't get it. The string is just going up and down.
Billy: And why is the display of the sine wave signal generator flashing?
Mr. p: Okay, first let's deal with the fact that the display is flashing.
Realize, LCDs or liquid crystal displays like this are not always on, the display actually flashes,
it's just that it generally happens too fast for our eyes to be able to see the flashing.
Here is what the liquid crystal display looks like slowed down 32 times slower than real speed.
Billy: Wow, the display is actually flashing, instead of always on. That, that is weird.
Mr. p: Okay, the flashing is pretty annoying,
so I am gonna apply some video effects to get rid of the flashing of the LCD...
Okay, let's watch what happens when I change the oscillator to 30 Hz...
And let's add a top view which is zoomed in to show the oscillator and the left half of the string.
Remember, the oscillator is now moving up and down 30 times every second.
Bobby: Uh, why isn't anything moving?
Bo: That is a good question.
Billy: Yeah.
Bo: Wait a second, look at the hanging mass in the lower right hand corner.
It's moving like this is a normal video.
Mr.p: The frame rate of this video is 30 frames per second.
Billy and Bo: Oh...
Bo:... Oh, so we are looking at 30 pictures of the demonstration every second
and the oscillator is going up and down 30 times every second.
Billy: That means the oscillator and string return back to the same location 30 times a second
and then the video camera takes a picture of it.
Bobby: That looks weird.
Mr.p: Which is why every video I show of the demonstrations from here on out
will be 32 times slower than real speed.
And, now you can see what is actually happening in the standing wave pattern.
Let's now put three different frequencies on the screen at the same time
to better see the differences between the three standing wave patterns.
Bo: Ya know, I still don't really get it. The string is just going up and down in all those examples.
Mr.p: Okay, well, how about this. I will use some video editing trickery to add an "echo" of each wave pattern
so you can better see the three standing wave patterns being created.
Bobby: Woah. There are like, different humps.
Billy: Yeah, 1, 2, and 3 humps, depending on the frequency.
Bo: Right, but the string is still just going up and down.
It's just there as a pattern showing how they go up and down.
Billy: Dare I say a "standing wave pattern"?
Mr.p: Absolutely Billy, these are the "standing wave patterns".
However, I want to remind you of what is going on here, which I can demonstrate by "plucking the string".
The wave pulses created by the oscillator are sent down the string, reflected and inverted,
and then sent back down the string.
The wave pulses going back and forth constructively and destructively interfere with one another
and set up the standing wave patterns.
You just do not see the wave pulses which are going back and forth,
all you see is the resulting standing wave pattern.
Which is why I have created this animation to show you the wave patterns
which constructively and destructively interfere to create the standing wave patterns.
Billy: That's cool!
Bobby: What is that?
Bo: I don't know...
Mr.p: Actually, let's look at the individual parts of the animation.
The blue periodic wave is moving to the right and represents the periodic wave
which is being generated by the wave oscillator.
The red wave is moving to the left and represents the periodic wave
which is the result of the blue wave being reflected and inverted by the fixed end on the right.
But the truth is that neither of those waves exists by themselves.
What we actually see is the constructive and destructive interference
of those two waves creating the black standing wave pattern.
Now let's pause the animation to discuss what happens at specific points.
With the waves paused here, Billy, tell me what you see.
Billy: At this point ... uh the red and blue waves have the same amplitude
but are mirror images of one another.
Bobby: Then we get total destructive interference.
Bo: Right, the red and blue waves completely cancel one another out and the result is just a flat string.
Mr.p: Absolutely, at this point there is total destructive interference of all waves
and the net result is no wave at all.
And Bobby, what do you see when we pause the animation here?
Bobby: Uh, well, the red wave is in the same location as the blue wave, right?
Mr. p: Correct.
Bobby: Then the two waves constructively interfere with one another and create the same wave shape
as the red and blue waves only with twice the amplitude, right?
Mr.p: That is correct Bobby.
At this moment, both waves constructively interfere with one another
and create the standing wave with twice the amplitude.
And Bo, what if we pause the animation here, which is somewhere
between total destructive interference and all constructive interference?
Bo: Okay sure so... Uh, if you pick any point along the hypothetical, yellow dotted line.
Mr.p: You can call that like the equilibrium position.
Bo: Right. So, if you pick any point along equilibrium, you can add the red and blue waveforms together
and see that they add up to the black standing wave pattern.
... I mean just look at it. Right here you can see that red and blue waves have the same amplitude
but are on opposite sides of equilibrium and therefore totally destructively interfere with one another,
which is why the black standing wave pattern is at equilibrium right there. ...
Right here you can see the two waves add up to the black standing wave pattern.
I mean really, it is true at every point along the line that the blue and red waves
add up to the black standing wave pattern.
Which, I guess, is the point, right?
Mr.p: Absolutely correct, Bo. Okay, I want to identify some specific locations on the standing wave pattern.
On this standing wave pattern there are seven locations of total destructive interference.
Those are called nodes.
There are also six locations between the nodes called antinodes
where the waves create a larger amplitude wave via constructive interference.
Bo: Nodes are locations of total destructive interference where the standing wave is always at equilibrium.
Bobby: And antinodes are locations where there is a large amplitude due to constructive interference.
Bo and Bobby: Got it!
Billy: Uh, you both just stole my thunder.
Bo and Bobby: Yeah
Mr.p: Okay, let's switch back to the real demonstration rather than the animation
and identify the locations of the nodes and antinodes.
Also, I have added 61 and 76 hertz as well, to show a total of 5 standing wave patterns.
Billy: ... Uh, I do not really see the standing wave patterns.
Bobby: Where are they?
Bo: I think I can see them.
Mr. p: Oh right, sorry. I'll add my video "echo" of each wave to make it easier to see the nodes and antinodes.
Is it easier to see them now?
Billly: Sure. 15 hertz has two nodes, one on each end and one antinode in the middle.
Bo: 30 hertz has three nodes. One on each end and one in the middle.
And two antinodes one between each pair of nodes.
Bobby: 45 hertz has four nodes. Again, one on each end and then two more nodes
which split the string into three parts where at the center of each part is an antinode.
Billy: I think I am sensing a pattern here. Starting at 15 hertz with one antinode
and two nodes. We add one antinode and one node for every 15 hertz we add to the frequency.
Mr.p: Yes it's pretty close to that. Although clearly it is not quite 15 hertz
because the standing wave pattern is set up at 61 and 76 hertz not 60 and 75 hertz.
Bobby: Wait. Do you mean the standing wave pattern will not work at other frequencies?
Mr.p: Yes. That is an important thing to realize Bobby.
Take a look at these two examples, which are at frequencies which are between the frequencies
where we were able to set up standing wave patterns.
I am showing you both with and without the "echo" video effect.
Do you see how there is no standing wave pattern being created at either of these two frequencies?
BBB: Yes. Sure. Yeah.
Mr.p: But why? Why is there no standing wave pattern being created at these frequencies?
Billy: ...Does it have to do with the fact that each end has to be a node?
Bobby: That's right. Each end is a fixed end and is therefore a node.
Bo: The standing wave pattern at 15 hertz represents half a wavelength.
Bobby: The pattern at 30 hertz represents one full wavelength.
Billy: Oh, oh, oh. Each standing wave pattern has to have an integer multiple of half a wavelength.
Bo: Right. Because each end has to be a node,
only integer multiples of half wavelengths will fit to create standing wave patterns along the string.
That makes sense.
Mr.p: Great job everybody.
In summary, standing wave patterns are waves generated in a medium which interfere
with waves which are reflected to create a pattern of waves which appear to stand in place.
Bobby: That's why they are called standing waves. Duh!
Mr.p: Nodes are locations where the wave interferences cause total destructive interference.
Antinodes are locations where the wave interferences cause constructive interference.
And standing wave patterns are only possible at specific wavelengths which create,
in the case of nodes on either end of the medium,
integer multiples of half wavelengths of the wave.
Thank you very much for learning with me today, I enjoyed learning with you.