This is the next video on waves. This is what we're gonna cover in this video. I want to
start this video by having you observe a pattern that's going to be very important to understanding
the rest of this video. I'm going to allow this wave to move for one full period of its motion,
and I'm going to observe how much displacement it covers in that one period of time. So one
full period will be the amount of time it takes this particle to move all the way
up to the top of the wave and then all the way back down to the very bottom. So let's
see how far the wave moves in that one period of time. Okay, this may be kind of intuitive,
but in one full period of motion the wave moved exactly one wavelength of length,
and there's no way that couldn't have happened. Like when you stop and think about it for a little
while, a wave has to move exactly one wavelength in one period of time. So this will always be true
for any wave that we deal with. Any existent wave at all in one full period of repetition,
a wave moves exactly one wavelength of displacement, so that means that we can come
up with a new equation for the velocity of a wave. We know that velocity is displacement over time,
and because the displacement the wave covered is its wavelength and the time it took is the period,
we can say that the velocity of any wave is its wavelength divided by its period,
and because period is equal to 1 over frequency I can rearrange this equation and say that the
velocity of a wave is always exactly equal to its wavelength times its frequency. This is going to
be a really important equation for problem solving in this unit. I'm going to give you some more
visual intuition about how this equation works. So again the velocity is the specific amount that
this wave will move in one second. The wavelength is the length of one complete pattern of a wave,
and the frequency is the number of wavelengths that will pass in one second, because frequency
means repetitions per second. So the number of full repetitions of waves that pass in one second
is the frequency. So to visualize this I can see that here I have a wavelength of 3 meters. I'm
going to imagine that we have a frequency of 1 Hz, 1 event per second. So this equation is telling me
that if I have this wavelength in this frequency, it's going to result in a velocity of 3 meters per
second. So let's now use this visual to see if that's true. I'm going to have a frequency of 1,
so that means that one wavelength will pass by in 1 second, and we'll see how much displacement
it covered. So 1 wavelength passed by. So I can see that that frequency was 1 Hz and the
wavelength was 3 meters, so that automatically implies that the velocity it was traveling was
3 meters per second. It traveled exactly 3 meters from where it started in that 1 second of travel.
We can imagine now that the frequency is 2, that 2 wavelengths will pass by and each wavelength is
3 meters long. So that means that the velocity is going to be 6 meters per second, and if I
allow this to happen, I allow 2 wavelengths to pass in 1 second, because that's the frequency,
I can see that yeah altogether that velocity was 6 meters. That traveled 6 meters in 1 second.
So the velocity is 6 meters per second. I can change the wavelength and imagine this is now
a wavelength of 1, and if I give that a frequency of 1 so that 1 wavelength passes by in 1 second,
I can see that the velocity was 1 meter per second and only traveled 1 meter in that 1 second.
Finally, we can imagine a frequency of 4. So 4 of these wavelengths are gonna pass in 1 second,
and when I do that I can see that the velocity is again the frequency times the wavelength. So again
that's 4 meters per second here. We can make 2 types of graphs of a wave: a displacement-distance
graph and a displacement-time graph. A displacement-distance graph shows
the displacement of the material that the wave is moving through for each distance along the
material. It's basically a 2D picture of the full wave. So as you can see here I have a wave with
a height from the very top to the bottom of 1.5 meters and a length of 4 meters. So if I were to
make a displacement-distance graph, the graph on the right would be what it would look like. I have
sections of the wave clearly marked off leading up to 4 meters. I can see that the amplitude is
going to be 0.75 meters because that's how far the highest point is from the mid point of the wave,
not from the very bottom. So this graph shows you the displacement of the material for any distance
along the material that you're looking at. We can use the displacement-distance graph to find
the amplitude of the wave and the wavelength of the wave. So the amplitude I can see is 0.75.
That's how far it is from the 0 point. And I can see that the wavelength is 2 meters because one
full repetition of that wave covers 2 meters. So we can use the displacement-distance graph to
find amplitude and wavelength. We can also make a displacement-time graph of the wave. This is
a little more complicated. A displacement-time graph shows the displacement of one particle in
the material that the wave is moving through on the y-axis and the time that has passed on the
x-axis. So I'm going to show you an example of making a displacement-time graph for this wave.
I'm going to only focus on this one point because the displacement-time graph only focuses on the
displacement of one particle in the material that the wave is moving through. So I'm going
to kind of ignore the rest of the wave and just focus on this one particle and what it's doing
as the wave moves through it. I've sectioned off 8 seconds on this graph. So I'm going to allow
the clock to run for 8 seconds and observe how the particle oscillates as the wave moves through it,
and for every moment in time I'm going to graph the vertical displacement of the particle on
the graph. So I'm going to make a red line that shows exactly how high up the particle is for any
one second of time. So just observe what happens when the wave begins to move through the material.
Okay so that itself produced a wave pattern, but remember this is not a physical depiction of
the full wave. This is a depiction of what one particle in the wave is doing for every moment
in time. We can use the displacement-time graph to find the period and frequency of
the wave as well as the amplitude. So I can see here that the period is 8 seconds. That's the
amount of time it took to complete one full wave pattern. So one full wave pattern on
the displacement-distance graph represents the wavelength, and one full wave pattern on the
displacement-time graph represents the period of the wave, and because frequency is 1 over period
we can always get the frequency as well. Here the frequency is 0.125 Hz, just 1/8 seconds,
and obviously the amplitude is still 0.75 meters. That didn't change.
We can combine info from both graphs to solve for the velocity of a wave. So because we can use the
displacement-distance graph to get the wavelength of the wave, and we can use the displacement-time
graph to get the period of the wave, we can combine these 2 pieces of information along
with the fact that frequency is 1 over period to find the velocity, because to find the velocity
you just need the wavelength and the frequency, so if you have a displacement-distance graph
and a displacement-time graph you can get the wavelength from the displacement-distance graph,
the time from the displacement-time graph, and put them together to solve for the velocity. So
here the velocity would be wavelength times frequency which is 2 meters times 0.125 Hz
which is equal to 0.25 m/s. So that is definitely the velocity of the wave,
and I can get that from the properties of these two graphs without observing the wave itself we
can also create graphs of motion given the other graph and the velocity I can imagine here that the
velocity of this wave is 0.33 m/s and figure out what the displacement time graph would look like,
because if the velocity is 0.33 meters per second and I can see that the wavelength of the wave is
2 meters from the displacement-distance graph, and the amplitude is 0.75 meters, plugging this
into my velocity equation and rearranging to get the frequency gets me a frequency of 0.165 Hz,
and the period is 1 over the frequency and when I divide that out I got 6 seconds.
So that means that this wave has a period of 6 seconds and an amplitude of 0.75 meters. So if
I draw that out on my graph this is what that looks like: an amplitude of 0.75 meters and
1 complete wave pattern in 6 seconds. So that's what you need to understand to use
displacement-distance graphs, displacement-time graphs, and calculate the velocity of a wave.