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# Properties of periodic waves

Video transcript

In the last video we talked
about the idea that if I start with some type of a string
there, and if I were to take the left end of the string-- I
could just have equally have done the right, but if I take
the left end of the string and jerk it up, then all the way
down, and then back to its resting position, it'll generate
this disturbance in the string. And the disturbance might
initially look like this after I've done that jerking
up and down once. And that disturbance is going to
propagate down the string. It's going to move down
the string like that. Let me color this in black. So this is right after I do that
first cycle-- that first jerking up and down. The string might look
something like that. And if we wait a little while,
the string might look something like this, assuming
that I only did that once. The string might look something
like this, where that pulse has actually
propagated down the string. That pulse has propagated
down the string. And in the last video, we said,
hey, this disturbance that's propagating down the
string, or propagating down this medium-- although it
doesn't necessarily have to have a medium- we called this
a wave. And in particular, this wave right here,
this is a pulse. This is a pulse wave because we
only have, essentially, one perturbation of the string. Now if I kept doing that-- if I
kept going up and down, and up and down, essentially, if I
periodically did it at regular intervals, then my
string would look something like this. Doing my best to
draw it neatly. It might look something like
this, where once again, the perturbations are going
towards-- the disturbances are going to move to-- the right. They're going to move to the
right with some velocity. And what I want to do in this
video is really focus on this type of wave. This type of wave
right here, which you can imagine, since I'm periodically
moving this left side up and down, up and down,
and creating these periodic movements in the wave, we call
this a periodic wave. This is a periodic wave. The motion is
repeated over and over again. So what I want to talk about is
some of the properties of a periodic wave. Now, the first
thing you might say is, hey, how far are you jerking
it up and down? How far are these movements
from rest? So if this is the resting
position right there, how far are these movements above the
resting position and below the resting position? And we call that the amplitude
of the wave. So that distance right there- I'll do it in
magenta-- that distance right there is the amplitude. Sometimes mariners will have
an idea of wave height. Wave height, they normally refer
to from the bottom-- from the trough-- of
a wave to its peak. Amplitude, we're talking
about from the resting position to it's peak. So let me label peak. I think you know what
peak means. Peak is the highest point on
the wave. That's the peak. And that's the trough. If you're in a fishing boat and
you wanted to see how big a wave is, you'd probably care
about the wave height-- not so much if your boat's sitting down
here, you have to care about this whole distance. But anyway, we won't talk
too much about that. So that's the first interesting
idea behind a wave. And not all waves are
being generated by Sal jiggling a string on
the left-hand side. But I think you get the idea
that these waves can represent many different-- this graph can
represent many different types of wave forms. And this,
essentially, displacement, from the resting position, or
from the zero position, that is your amplitude. Now the next question you might
ask is, OK, I know how far you're jiggling this string
up and down, but how quickly are you doing it? So how long does it take for you
to go all the way up, all the way down, and back again? So how long for each cycle? Cycle is me going up,
down, back again. How long for each cycle? Or you might say how long
for each period? We're saying this is periodic. Each period is each repetition
of the wave. So this idea of how long for each cycle, we
call that the period. And this is going to
be a unit of time. Maybe I'm doing it every
two seconds. It takes me two seconds to
go up, down, back again. Up, down, back again. That's going to be
two seconds. A very related term is,
how many cycles am I doing per second? So in other words, you
could say, how many seconds for each cycle? We could even write that. So for example, a period might
look like something like 5 seconds per cycle. Or maybe it's 2 seconds
per cycle. But what if we're asked how
many cycles per second? So we're asking the
opposite question. It's not how long, how many
seconds does it take for me to go up, down, and back again. We're saying in each second, how
many times am I going up, down, back again? So how many cycles per second? That's the inverse of period. So period, the notation
is normally a big capital T for period. This is frequency. It's normally denoted by an F. And this, you're going to
say cycles per second. So if you're going 5 seconds per
cycle, that means you're doing 1/5 of a cycle or, 1/5
of a cycle per second. All I did is invert
this right there. And that make sense. Because the period and the
frequency are inversions of each other. This is how many seconds
per cycle. How long does one up, down,
back again take? And this is how many up,
down, back agains are there in a second? So they are inverses
of each other. So we could say that frequency
is equal to 1 over the period. Or you could say that
period is equal to 1 over the frequency. So if I told you that I'm
vibrating the left end of this rope at 10 cycles per second--
and, by the way, the unit of cycles per second, this is a
hertz, so I could have also written this down as 10
hertz, which you've probably heard before. 10 hertz just means 10
cycles per second. If my frequency is 10 cycles per
second, my period is going to be 1 over that. So 1 over 10 seconds per cycle,
which makes sense. If in 10 times, I can go up and
down, a whole up, down, back again, if I can do that
10 times in a second, it's going to take me 1/10 of a
second to do it each time. Now another question we might
ask ourselves is, how quickly is this wave moving, in this
case, to the right? Since I'm jiggling the left
end of the string. How quickly is it moving
to the right? So the velocity. So to do that, we need to figure
out how far did the wave go after one cycle? Or after one period? So after I jiggled this once,
how far did the wave go? What is this distance from this
resting point to this resting point there? And we call that a wavelength. And there's a lot of different
ways you can define a wavelength. You could view a wavelength as
how far the initial pulse went after completing exactly
one cycle. Or you could view it as the
distance from one peak to another peak. That is also going to
be the wavelength. Or you could view it as a
distance from one trough to the other trough. That's also the wavelength. Or in general, you could view
the wavelength as one exactly equal point on the
wave. From that distance to that distance. That is also one wavelength. Where you're completing, between
that point and that point, you're completing one
entire cycle to get exactly back to that same point. And when I say exactly back to
that same point, this point doesn't count. Because this point, although
we're in the same position, we're now going down. We want to go to
the point where we're in the same position. And notice over here,
we're going up. We want to be going up again. So distance is not
one wavelength. To go one wavelength,
we have to go back to the same position. And we're moving in the
same direction. So this is also one
wavelength. So if we know how far we've
travelled after one period-- let me write it this way;
wavelength is equal to how far the wave has traveled
after one period. Or you could say after
one cycle. Because remember, a period is
how long does it take to complete one cycle. One to complete up, down,
and back again notion. So if we know how far we've
traveled, and we know how long it took us, it took us one
period, how can we figure out the velocity? Well, the velocity is equal to
distance divided by time. For a wave, your velocity--
and I could write it as a vector, but I think you
get the general idea. Your velocity-- what's
the distance you travel in a period? Well, the distance you travel in
a period is your wavelength after one up, down,
back again. The wave pulse would have
traveled exactly that far. That would be my wavelength. So I've traveled the distance of
a wavelength, and how long did it take me to travel
that distance? Well, it took me a period
to travel that distance. So it's wavelength divided
by period. Now I just said that 1 over the
period is the same thing as the frequency. So I could rewrite this
as wavelength. And actually, I should
be clear here. The notation for wavelength
tends to be the Greek letter lambda. So we could say velocity
is equal to wavelength over period. Which is the same thing
as wavelength times 1 over my period. And we just said that 1 over the
period, this is the same thing is your frequency. So velocity is equal to wavelength times your frequency. And if you know this, you can
pretty much solve all of the basic problems that you might
encounter in waves. So for example, if someone
tells you that I have a velocity of-- I don't know-- 100
meters per second to the right, so in that direction--
velocity you have to give a direction-- and they were to
tell you that my frequency is equal to-- let's say my
frequency is 20 cycles per second, which is the same
thing as 20 hertz. So literally, if you had a
little window where you're only able to observe this part
of your wave, you'd only observe that part
of my string. If we're talking about 20 hertz,
then in 1 second, you would see this go up and
down twenty times. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
11, 12, 13, 14, 15, 16, 17, 18, 19, 20. In exactly one second,
you would see this go up and down 20 times. That's what we mean by the
frequency being 20 hertz, or 20 cycles per second. So, they gave you
the velocity. They gave you the frequency. What's the wavelength here? So the wavelength, in this
situation-- you would say the velocity-- your velocity is
equal to your wavelength times your frequency. Divide both sides by 20. And actually, let me make sure
I get the units right. So this is meters per second,
is equal to lambda times 20 cycles per second. So if you divide both sides by
20 cycles per second, you get 100 meters per second times
1/20 seconds per cycle. And then this becomes a 5. This becomes a 1. So you get 5, and then the
seconds cancel out. So you get 5 meters per cycle. So this is equal to 5 meters per
cycle, which would be your wavelength in this situation. So it's 5 meters. You could say 5 meters per
cycle, but wavelength implies that you're talking about
the distance per cycle. So in this situation, if this is
moving to the right at 100 meters per second and this
frequency-- I see this moving up and down 20 times in a
second-- then this distance, right here, must be 5 meters. Likewise, we can figure out
the period very easily. The period here is just going
to be 1 over the frequency. It's going to be 1/20
seconds per cycle. So using these formulas-- and I
don't want you to memorize a formula, it should be
intuitive for you. And hopefully, this video made
it a little bit intuitive. But using this, you can really
answer almost any question if you're given two of these
variables and you need to solve for the third. Anyway, hopefully you
found that helpful.