Hi. It’s Mr. Andersen. And this is AP Physics essentials video 116. It is on harmonics.
Harmonics are integers of the fundamental frequency. That is quite a mouthful. What
does that actually mean? Well the example of the guitar might help a little bit. So
if we look at a guitar there are some frets on it. And you will notice there are some
markings on the frets. But there is one special fret. It is called the twelfth fret. And so
what does that mean? The distance to here, from here to here is about 2 feet. But if
I take that distance times two I get right to the end. And so a fundamental frequency
is just plucking a string and its the whole length of that vibrating string. We call it
the fundamental frequency. And so we could look at it like this. This would be the fundamental
frequency at one end and the other end we just get these giant vibrations, these big,
what are called, standing waves. This is the first harmonic. The second harmonic would
look like this. The second harmonic is going to have a node at the end and a node at the
other end, but also a node here in the middle. And you will notice that that comes right
on the twelfth threat, half way between the two. We could also have the fourth harmonic
that is going to be at the twelfth fret as well. And so if you really know how to play
guitar you can hold just briefly on the twelfth fret and as you strum it you are not just
going to have a fundamental frequency but you are going to have all these frequencies
on top of themselves. And this is what a skilled guitarist sounds like when they are playing
these harmonics. Pretty cool. So standing waves remember are created when we have interference
of waves going up and down and up and down. So that interference pattern creates these
standing waves. Waves that appear to not move. Now if we have standing waves the wavelength
is determined by two things. First of all it is determined by the boundary, in other
words the ends of that string if we are talking about a guitar. Or it could be the ends of
a tuba if we are talking about some kind of a wind instrument. Or its frequency, that
is the other thing that can affect the wavelength, in other words how fast we are vibrating the
air inside it. And so let’s give you an example of that. Let’s say we are looking
at the fundamental frequency for a given boundary L, or given length of L. The fundamental frequency
is going to be the distance where we can have a portion of a wavelength fit inside there.
Now if we think about this, how many wavelengths are in this boundary? Remember a wave goes
up to the crest and then it has to go down to the trough. And so how many wavelengths
would the fundamental frequency be fitting in this boundary? It is a half. And so really
if we look at what is the wavelength, the wavelength is going to equal 2 times the length.
And so based on that boundary we know immediately what is going to be the wavelength of that
fundamental frequency. Now that we have that length, in other words it two times the length
of the boundary, then we can plug that in to figure out frequencies. The frequency is
the velocity of the wave divided by the wavelength. Remember, because velocity always equals frequency
times wavelength. And so we are solving for frequency. And so what I can do is I can simply
take this value, 2 times the length of the boundary. I can plug that in here for wavelength.
I know the velocity or the speed inside that string or inside that chamber. And then I
can solve for that frequency. Now what we have identified is the length and the frequency
of the fundamental frequency. We call that the first harmonic. What would a second harmonic
look like? It is going to look like this. And so a second harmonic, we can have an entire
wave. Look, we go to the crest and then we go the trough and then back again. And so
in this case what is going to be the wavelength? The wavelength is going to equal the boundary
because we could put one whole wavelength inside there. Once we know this, then we could
figure out the frequency of the second harmonic. And then we could figure out the third harmonic.
The third harmonic, here is a whole wavelength here. It goes up. And then down. And so it
would actually be, if we were looking at the length, it is going to be two-thirds the length.
And so what is neat once we know the boundary we could figure out the wavelength. Once we
know the wavelength, if we know the speed, then we can figure out the frequency. And
so let’s apply that a little bit. So let’s say you are given a problem where you say
we have string. It is 2 meters long. And you need to calculate what is going to be the
wavelength. Well that is pretty easy. We say that 2 lengths equals one wavelength. In other
words remember we have to go up and then we have to go down again. And since L is 2, what
is going to be my wavelength? I simply multiply it. So it is 4.0 meters, using significant
digits. So now let’s go to the next one. So this would be the second harmonic. Wavelength
of the second harmonic equals L. So what is going to be my length? My length now is going
to be 2.0 meters. You can see that the wavelength is getting shorter. If we keep going like
this, so this would be the third harmonic. So our wavelength equals two-thirds the length.
I could solve for my wavelength, 1.3 meters. And I could just keep going like that. You
can see here for the fourth harmonic, that would be 1.0 wavelength. And we are half way
to the length. And so it is going to be like that. And so that would be the wavelength.
Now, once we plug that wavelength in you can see that as we decreasing the wavelength,
what is going to happen to the frequency? The frequency is going to have to go up. And
so these are all the values we would have for the first harmonic. You just figure out
what harmonic are we doing. How many wavelengths would fit in. So let’s say if we are looking
at this one right here, here is one wavelength. So that wavelength, two-sevenths of the length.
And then we could solve for the frequency. #00:05:38-6#
Let’s get to calculating the wavelength and calculating the frequency. We are using
a simbucket simulation. So we have a really long string. It is 7.5 meters long. It is
almost like 30 feet long. And so it is going to move back and forth like that. You can
see in the simulation that I have covered up the wavelength. You can not see what that
is, the period and the frequency. So we are going to try to figure out first of all what
is the wavelength. Once we have that, let’s figure out the frequency. And so what harmonic
is this? Well we have nodes on either side. And so we know that this is the first harmonic.
And so what is our wavelength? 2 L. And so now we just solve for that. So we are taking
2 times 7.5, which is the length. And so what is the wavelength of the fundamental frequency
of the first harmonic? It is going to be 15 meters. Let’s check in the simulation to
make sure that is right. Yeah. That is in centimeters. But 15 meters is right. Now that
we have figured out the wavelength has 15 meters, let’s figure out what the frequency
is. Frequency remember equals the velocity or the speed divided by the wavelength. And
in the simulation we are given the speed as 100 centimeters per second or that would be
be 1 meter per second. So it is 1 meter per second on the top. What is our wavelength?
15 meters on the bottom. And so what is going to be our frequency? 0.67 Hertz. And so if
we look at the simulation, that is right. Now how could we go from here to figure out
the period? Remember the reciprocal of that, 1 divided by the frequency is going to give
you the wavelength. If you were to try one on your own, so first of all you have to figure
out what harmonic this would be. So we have, how many wavelengths fit in there? So this
would be the second harmonic. How many wavelengths fits in? Just one. So we know that the wavelength
equals the length of the boundary. So we have that. Now that we know that the whole length
is 7.5 we could figure out that the wavelength is 7.5 as well. And now if we want to figure
out the frequency, what is the frequency going to be? Well we have the same speed through
the string or through the wire. So our velocity is going to be the same. It is going to be
1 divided by the 7.5, which is our wavelength again. Our wavelength at this point has gotten
smaller. So what is that going to do to our frequency? Our frequency is actually going
to increase. So if we check that out, 0.13 Hertz, our frequency. And then we are going
to have period that is less. And so you can see that as we move up in harmonics, what
happens to the wavelength? It becomes smaller. And as a result what happens to the frequency?
The frequency is going to go up. So did you learn to understand what harmonics are and
how to calculate them based as an integer value of the fundamental frequency? And then
finally are you able to calculate the wavelength and then the frequency for a given standing wave?
I hope so. And I hope that was helpful.