Mr:p: Good Morning.
In a previous lesson we showed standing wave patterns on a strings and
explained that standing wave pattern are only able to be created
at specific wavelengths and therefore frequencies.
For example, standing waves are possible at 15, 30, and 45 hertz on this string.
However, while working on the demonstration,
I noticed that I was able to set up a standing wave pattern at 22 hertz.
[intro]
So here's the question.
If standing waves are only "allowed" at 15 and 30 hertz and nowhere in between on this string,
then why do we see a standing wave pattern at 22 hertz?
Bo: Nope. I don't see the standing wave pattern.
Bobby: Not at 22 hertz. I see it at 15 and 30 hertz though.
Billly: Yeah
Mr. p: Okay. I'll add to the video "echo" effect and you should be able to see it better.
Billy: I see it now
Bo: I can see it, however , the standing wave patterns at 15 and 30 hertz clearly have larger amplitudes, right?
Bobby: I agree.
Mr.p: Okay, yes, the standing wave patterns at 15 and 30 hertz have larger amplitudes, however,
there is still a standing wave pattern being created at 22 hertz.
Let me add a zoomed in view of the 22 hertz example showing the left half of the string
with both the slow motion and the video "echo" effect as well.
Now, I want to get back to the fact that a standing wave previous pattern exists here,
even though we determined in a previous lesson that this was not possible.
So, I want to know, we said it is not possible, but there it is.
Why?
BBB: I don't know. Ya got me. Me either.
Mr. p: Oh come on. Y'all can do better than that.
Billy: Well, why did we say it was not possible in the first place?
Bo.: Because there are nodes, or locations of total destructive interference on either end of the string.
So, the only wavelengths which are possible on this string have to end with the wave
at the equilibrium position at both ends of the string.
For example, 15 hertz has half of a wavelength, 30 hertz has 2 half wavelengths or one full wave,
and 45 hertz has three half wavelengths or one one half wavelengths.
Each of those is an integer multiple of a half wavelength
and therefore both ends of the wave are at the equilibrium position.
Bobby:...Well that's it then.
Bo: What?
Billy: Yeah, What?
Bobby: Look closely at the left end of the string where the oscillator is ....
where the oscillator is going up and down 22 times every second.
Billy: Oh, and that end is not really a node!
Bo: Yeah, it's close to a node because it only oscillates up and down a little bit, however, it's still not a node.
Mr.p: Right. The left end is not actually a node, so, in this example,
we can get wavelengths which are not integer multiples of half wavelengths.
...Billy, How many nodes are in this standing wave pattern?
Billy: How many nodes? So how many locations have total destructive interference?
Well, the the right end is a node because it really is fixed in place
and it it looks like there is one
other node a little more than a third of the length of the string to the right on the oscillator
Correct. There are two nodes in the standing wave pattern at 22 Hertz
Also, we know the length of the string is 0.865 meters from a previous lesson
And I measured the distance from the oscillator to the node that is nearest to the oscillator
to be 0.265 meters...
Billy: That means we can determine the wavelength of the wave in this standing wave pattern!
Bobby: And once we know that we can determine the speed of the wave.
Billy: Okay, there is half a wavelength between the two nodes on the string
and we know that that distance is the length of the string minus 0.265 meters or 0.60 meters.
So the wavelength of the standing wave pattern on this string at 22 Hertz is, well, 1.2 meters.
Bobby: And we know the speed of the wave equals frequency times wavelength or
22 times 1.2 which equals 26.4, or 26 meters with two significant digits.,
Bo: And in the previous lesson, we determined the speed of this wave to be 26.1922
or 26 meters per second with two significant digits.
I would say that confirms our hypothesis of why this standing wave pattern can exist on this string. Cool.
Billy: yeah cool.
Bobby: Nice.
Mr.p: Absolutely, you figured it out and showed the wavelength and frequency
result in a similar wave speed to a previous video. Very nice.
Now I do want to reiterate that
the only reason this standing wave pattern was able to be set up on this string
between the allowable half wave lengths
is because one of the two ends of the string was not quite a node.
This is NOT true of examples where the two ends have to be nodes like a stringed instrument or
where the two ends have to be antinodes like an open pipe instrument or
where one end has to be a node and the other end has to be an antinode like a closed pipe instrument.
Billy: What are those?
Do not worry Billy, we will get to those in future lessons.
Thank you very much for learning with me today, I enjoyed learning with you.
Billy: But I want to know now. Mr.p:I know Billy.