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visit MIT OpenCourseWare at ocw.mit.edu. YEN-JIE LEE: Hello, everybody. Welcome back to 8.03. Today we are going to continue
the discussion of waves. We will discuss a very
interesting phenomenon today, which is dispersion. And before that, we
will discuss a bit, just to give you some
reminder, about what we have learned so far. So we discovered
this wave equation, which is showing
here, in the class, and then we also show
you that it described three different kinds
of systems, which we included in the lecture-- the massive strings,
which are the strings can actually oscillate up
and down in a wide direction. And also we discussed
about sound waves. This is also discussed
in a previous lecture. And sound waves can be
described by wave equation. And finally, the last
time we discussed electromagnetic waves. It's a special kind
of wave involving two oscillating fields. One is actually
the electric field, the other one is magnetic field. So that's kind of interesting,
because this is actually slightly different
from what we've discussed before in
the previous two cases. This is actually a
three dimensional wave, and also involving two
different components. And we also discussed
the solution, the traveling wave solution
of the electromagnetic waves. As you can see from
here, the electric field is showing us the red,
and the magnetic field is showing us the blue. And you can see that in
case of traveling wave, they are in phase. And the magnitude reach
maxima simultaneously for electric field and
the magnetic field. And while in the case
of standing wave, there's a phase difference.
so they don't reach maxima simultaneously in the standing
electromagnetic field case. OK, so what are we
going to discuss today? We would like to discuss the
strategy to send information using waves. How do we actually send
information using waves? So you can say, OK,
maybe I can just send a harmonic oscillation. So If I do this
harmonic oscillation, I can basically
produce harmonic waves. They are moving up and
down, and is actually always constant angular
momentum and angular frequency. And maybe that's a way
to send the information. But this kind of wave is, in
reality, not super helpful, because if you fill the whole
space with harmonic waves, then you don't know when did you
actually send the signal. Because it's always
oscillating up and down, so you don't know the
starting time of the signal. So in reality, these kind of
simple harmonic oscillating traveling wave is
not super helpful. So what is actually helpful? That's the question. So what is actually helpful
is to produce square pulse, for example. We can create square pulse,
for example, in this case, I can create a
square pulse here. And in the next time interval,
I don't create a square pulse. In the next time interval,
I don't do anything. And I create another square
pulse here, et cetera, et cetera, OK. If you use this kind of
strategy what we can do is to have some kind of receiver
here to actually measure the magnitude of the pulse. And then we can actually
interpret this data. So this wave is going to
where the positive x direction or going to the right-hand
side of the board. And the receiver will be
able to interpret this data by appraising this
ratio on the energy or on the measure
of the amplitude. Then I can say, oh,
now I'd receive a 0, and then the next
signal I'm receiving is 1, and this one is
0, and 0, and 1, and 0. In this way, I can
actually send information and that this information can be
verified as a function of time. So in short, what
would be useful is probably to use a
narrow square pulse, and that would be very helpful
in transmitting information. So if we consider an
ideal string case-- if I have an ideal string,
as we learned before, the behavior of the string is
described by the wave equation. Partial squared psi
partial t squared, and this is equal to v
squared partial squared psi partial t squared. And this v is actually
related to the speed of the progressing
wave, as we discussed before-- the progressing
wave solution. And if I have this
idealized string, and it obey the wave equation,
the simple version of wave equation, then I would be
able to divide the dispersion relation. So I can now write down my
harmonic progressing wave in the form of sine
kx minus omega t. If I have a harmonic
oscillating wave propagating toward the positive x
direction at the speed of v. I can write it down in
this functional form, where k, as a reminder,
is the wavenumber, and the omega is actually
the angular frequency. And therefore, if I plug in
this solution, and of course, it can have arbitrary amplitude. If I plug in this
solution to this equation, then what I'm going
to get is, as we did in the last
few lectures, there would be a fixed
relation between k, which is the wavenumber, and the
omega, the angular frequency. So the fixed relation
is actually omega over k would be equal to
v, which is actually the velocity in
this wave equation. And from the
previous discussion, we know this is actually
equal to a squared root of T over rho L, where T is
actually the tension, the constant tension, which
we apply on this string, and the rho L is actually the
mass per unit as a reminder. So what does this mean? What does this equation mean? We call it dispersion
relation a lot of time, right? But we actually didn't
explain why do I do that. So we are going to learn
why this is actually called this dispersion relation. Omega is a function of k. And in this case, in this very
simplified idealized case, omega over k is ratio. we know this is related to
the speed of propagation of the harmonic wave is equal
to v. v is a constant that is independent of k. This ratio is independent of k. What does that mean? That means if I prepare waves
with different wavenumber, or in other words, waves
with different wavelengths, they are going to propagate
at the same speed. So the speed of the
harmonic progressing wave is independent of
the wavelength. That's actually
very good, because in this case, if I
prepared the square pulse, as we learned before, this
square pulse is actually a very complicated object. Square pulse is really
very complicated. You can do a Fourier
decomposition as we did before. And we need infinite
number of turns of harmonic oscillating waves. We actually add them
together so that I can produce a square pulse. And as I mentioned here, if
the dispersion relation, omega over k, is is our constant,
v. That means all the whatever wavelengths pulse,
which should be added together and produce the
square pulse, are going to be traveling
at the that speed. Therefore, if I have this
square pulse in the beginning, after some time, t,
what I'm going to get is that this is the original
position of the square pulse, and after some time,
t, this square pulse will move by v times t in
the horizontal direction. And the shape of
the pulse is not going to be changed, because no
matter what kind of wavelengths which produce the
square pulse, all the components in
the square pulse are propagating
at the same speed. So this kind of system,
which has satisfied this kind of dispersion relation
is called nondispersive media. no dispersion was
happening in this case, in this highly idealized case. We also know that in
case of the string, we are actually making
it too idealized. So if we consider a
more realistic string, then I have to consider an
important phenomenon, which is-- or is a important property
of the string, for example-- stiffness What do I mean by stiffness? So for example, if I take
a string from a piano, a piano string, even if
I don't apply any tension to the string, if
I bend the string, it don't like it, all right? It's going to bounce back and
restore to its original shape. So that's what I call stiffness. It's a different contribution
compared to the string tension. So what we have been discussing
so far that this distorting force is actually coming
from the string tension, t. OK? What will happen if I introduce
additional contribution from the stiffness? The stiffness is actually
not completely related to the string
tension, and that also wants to restore the
shape of the string. OK? Before we go to the
modeling, I would like to take some
votes to predict what is going to happen. How many of you were predict
that if I introduce and include the stiffness of the
string into my equation, will the speed of
propagation increase? How many of you think
it's going to happen? 1, 2, 3, 4, 5. OK. So some of you predict the speed
of propagation will increase. How many of you
predict that the speed of propagation of the harmonic
wave will stay the same? How many of you? One? OK, only one. OK, how many of you
actually predict that the speed of
propagation would decrease OK so all the other students
don't have opinion. OK, want to wait for the answer. All right. So you can see that it is
actually not completely obvious before we solve this question. And we are going to solve it
with a simple model, which actually slightly modifies
the idealized wave equation. So now, one semi-realistic
model which I can introduce is to add a term additional
term to the wave equation. So I can now rewrite
my wave equation to include the effect that to
describe a realistic string, and now this is your partial
squared psi partial t squared. This will be equal to v
squared partial squared psi partial t squared. And the additional term,
which I put into this, again, is minus alpha partial to
the 4 psi partial x to the 4. And this is actually
the contribution from the stiffness. This is stiffness. OK, so you can see that the
wave equation is now modified. And what I could do in order to
get the relation between omega and the k-- what I could do
is that I can now start with this harmonic wave
solution progressing wave solution, plug that in to
this equation, this modified equation, and see
what will happen. If I plug this
equation into it that modified wave equation,
what I am going to get is the following. So basically the left-hand, side
you're going to get minus omega squared. And then the
right-hand side, you get v squared minus k squared
and plus alpha k to the 4 in the right-hand side. OK, so of course, I can
now cancel this minus sign. This will become plus and
this will become minus. And then you can see that
the relation between omega and the k is now different after
I introduce this term, which is proportional to alpha. Alpha is actually describing
how stiff this string is. Of course, now I
can calculate omega over k, which is
actually, as we learned before, right is the
speed of the propagation of a harmonic wave. So basically, if
I calculate omega over k from this equation,
then basically what you get is v square root of 1
plus alpha k squared. So if you look at this
equation, the first reaction is, oh, now this
omega and the k ratio is not a constant anymore
as a function of k. What does that mean? That means if I prepare
progressing waves with different wavelengths
for wavenumber k, it's going to be propagating
at different speed, OK? Before we introduce
this into the model, the ratio omega and k is a
constant v, independent of k. Now, once you introduce this
model into the equation, and you plug in the
progressing wave solution to actually check the
dispersion relation obtained from this equation, you'll find
that the speed of progressing wave depends on how distorted
this progressing wave is, OK? So let me compare
this to situation in this graph, omega versus k. So we will see this dispersion
relation graph pretty often in the class today. The y-axis is actually the
omega, angular frequency, and the k is the wavenumber,
two pi over lambda. OK. In the original
case, in the case I have this idealized
string, obey the wave equation
which we introduced in the previous lectures. If I plug omega as a function
of k, what I'm getting is a straight line. question. AUDIENCE: Why are
you [INAUDIBLE] minus alpha [INAUDIBLE]. YEN-JIE LEE: This one, right? AUDIENCE: [INAUDIBLE] YEN-JIE LEE: Oh, maybe I
made some mistake here. So this should be
also plus here, right. So you have this-- OK, so this is omega
squared, and I shouldn't have this minus sign here, right? So this should be minus,
and this should be-- OK, let's go back to the
original equation, OK. So basically, you get-- so if I plug in this
equation to this equation, so basically I get minus
omega squared out of it. And I get minus k
squared out of this. And I'm going to get
plus k to the 4 out of this partial square to
the 4 psi partial x to the 4. Therefore, this would be minus. OK, maybe I made a mistake. Thank you very much
for spotting that. AUDIENCE: [INAUDIBLE] YEN-JIE LEE: Oh,
yeah, I'm sorry. Not my best day today. AUDIENCE: [INAUDIBLE] YEN-JIE LEE: Yeah. Well, then I do it. OK. I must be drunk today. [LAUGHTER] Thank you very much. Anymore mistake? OK. Fortunately not, right? OK. Very good. So let me do this again. So now I can modify my
wave equation, right? Originally, the wave
equation is partial squared psi partial t squared equal to
v squared partial squared psi partial x squared. And now I add additional
term, which is actually proportional to the partial to
the 4 psi partial x to the 4. OK, if I add this
term into again. And now I plug in the wave
equation, the progressing wave solution, into this
equation, and I would get this formula, OK? So now everything
should be correct, and I have clear evidence
that everybody's following. So that is very good. And now, I can now cancel
all the minus sign, right and then it's become plus. And now I can actually
calculate what would be the speed
of propagation for this specific
harmonic progressing wave and omega over k will be
equal to v square root of 1 plus alpha k squared. OK? Thank you very much
for the contribution. And then now we see that
here this ratio depends on k. So if I plug this on top of
the previous curve, which is actually obtained from here,
then what I'm going to get is something like this. In the beginning
it's pretty close to the nondispersive case. And it goes up, because of
this alpha contribution. Alpha is actually a
positive number in my model. And the k is actually
the wavenumber. So what is going to happen
is that basically after you include stiffness, the
slope of this curve is changing as a function of k. OK? What do I learn
from this exercise is that if I increase k,
if I have a very large k-- that means I have a very small
lambda, because k is actually 2 pi over lambda. OK? So that means I'm looking at
something really distorted like this. Both string tension
and the stiffness wants to restore the
string back to normal. Therefore, what is
happening is that you are going to get
additional restoring force. Therefore, as we actually
calculate to here if alpha is actually positive,
then the velocity actually increased with respect
to what we actually get before we actually
had this into a model. So I think that makes sense,
because the stiffness also wants to restore the distortion. Therefore, you have larger
and larger restoring force. Therefore, the
speed of propagation of this harmonic
wave will increase. so that's pretty nice. But what does that
mean to our project? OK, our project is
to send information from one place to the
other place, right? So what we just discussed
is that we can actually send a square pulse
and let it propagate. A square pulse can be decomposed
into many, many pieces-- many, many harmonic waves. OK? Before the square pulse
works, because all the waves with different
wavelengths should be moving at this constant
speed, independent of the wavelengths. Now we are in trouble. As you can see here, now the
speed, which is omega over k depends on the
wavenumber or wavelength. Therefore, different
components, which actually are needed to
create a square pulse, are going to be propagating
at different speed. You can say, oh, come on,
this is actually mathematics, so I don't believe you. A square pulse is
a square pulse, and that's mathematics,
that's math department. But we can actually really
see that in the experiment. OK? So that's-- take a look
at this demonstration. Maybe you didn't
notice that before, but we have seen this effect
from the previous lectures. OK, so I can now
create a square-- not really a square pulse, but
actually some kind of pulse. OK I can create some
kind of pulse like this. OK? And as we learned before,
when this pulse pass through an open end, it's
going to be bounced back. so therefore, I can have-- I can actually show you this
demo in a limited set-up. But this pulse is going to be
going back and forth, because I have open end, as
we've discussed before. What is going to happen
is that since we have a realistic system,
what is going to happen is that this pulse will
become wider and wider, right? That's the prediction
coming from this equation. Different component with
different wavelengths is going to be propagating
at different speed. Therefore this pulse is
going to become wider, and we can see that. OK, so let me quickly produce a
pulse and see what will happen. OK. Originally, it's
actually really sharp. And you can see that really
the width of the pulse become wider and wider. And at some point, it disappear. If I have a very long set-up,
what you are going to see is that it's going
to be propagating toward the same direction. And the width of the
pulse is actually going to be increasing
as a function of time. Let's take a look at this again. Now, this time we
have a negative pulse. You sort of see-- very similar, see. And also you can
see that there are some strange vibration actually
left behind the main pulse. So that means harmonic waves
with different wavelengths really propagating
at different speed. And for that, to
demonstrate this effect, I also prepared
some demonstration, which actually are based
on our calculation, OK. So you can say that, OK, now I'm
convinced I can see dispersion in the experiment. How do I know this
calculation actually match with the experimental
data, right? How about we really run
a simulation and see what would happen. So what this example actually
do is, in the beginning, you would do
integration like crazy in order to get all the
components calculated. Then it's going to
propagate all those pulses-- all those pulse with different
components through the medium, OK? And then there will be
two different colors, one is actually blue, which
is the original shape. The other one is
actually the one which is stiffness turned up. So now, in the beginning
I can set the alpha value to be 0.02 and see
what will happen. And I will put produce
triangular pulse. You can see that now. The program is really
working very hard to capture all the components
from 1 to 199 and equal to 1 until 99. And then now, these
individual components are propagating
through the medium. And you can see that
originally the shape is like-- the blue shape--
triangular shape. And you can see that
is a function of time. The pulse become
wider and wider, OK? Now, of course, I can
increase the alpha to 0.02 and see what happen-- from 0.02 to 0.2 and
see what will happen. You should expect a
much larger dispersion. And you can see that
now in the beginning, it's doing the integration. And you can see that this
time because the alpha is actually larger. Therefore, you see that this
effect, this broadening, is actually happening
earlier, and it become broader and
broader, and that there are a lot of strange
structures, as you can see also from the demo, produce because
different components are actually propagating
at different speeds. So of course, we are MIT, so
in this course we have MIT-- MIT waves. So let's take a look at the MIT
wave and see what will happen. Now you see that there
are very sharp edge, which actually require really a lot
of effort to reproduce that. And you can see that
MIT is kind of distorted as a function of time. We can kind of still
identify the peak, but it's actually now displaced. And in the end of
the simulation, you can not even
recognize that's actually originally MIT signal, which
was sent from your source. So what I want to say is that
this effect, this dispersion effect, is really
an enemy, which is actually very dangerous. And that actually
will prevent us from sending high
quality signals. OK, any questions
about all those demos? Yes. AUDIENCE: Why do we
model the [INAUDIBLE]?? YEN-JIE LEE: So this is because
the stiffness is actually symmetric, right. So if you bend the
string, then there are contribution from the
positive and negative part, OK? If you have partial to the
3, partial to the x to the 3 component, then it's going to
be a symmetric and so actually against our physics intuition. And also, in this
modeling, you also match with our experimental
data pretty well. OK. Very good question. And on the other hand, we now
consider then the stiffness. you can also go back to
the infinite number coupled oscillator case. If you instead take an
example which is actually not super small displacement
approximation, you take the next to
leading order term. Then you will see that the
partial to the 3 partial x to the 3 term as you cancel
because it's symmetric, or so I argued. And then you will be able to
also obtain this tern when you have slightly larger
displacement with respect to the equilibrium position. So I hope that
answers your question. Any other question? Yes? AUDIENCE: If you were looking at
[INAUDIBLE],, for example, what would be [INAUDIBLE]? YEN-JIE LEE: When you
pass through the medium. AUDIENCE: So [INAUDIBLE] YEN-JIE LEE: A
molecule can actually change the speed of
different wavelengths, actually, differently, right? Very good question. OK, so very good. We got two questions, and
we can see that if I now turn on the alpha and make
the alpha value large, then you can see that the
information is distorted. And this involve
infinite number of terms. And in this case, in this
new demo which I show here, I have alpha value equal to 0.2. Therefore, the
effect of dispersion is actually much larger
than what you showed before. And then you can see that
this MIT wave quickly become something like a
Gaussian-like wave, right? OK, so very good. So you can say, OK, you
are making an example-- it's a very interesting example,
but it involve too many terms. You have infinite number
of progressing waves in this example. It's very difficult
to understand. How about we go back to a
much simpler example, OK? What we can do is
that instead of going through infinite number
of harmonic waves, now we just consider two waves,
and overlap these two waves together and
see what will happen. And let's see what
we can learn from it, because the required number of
harmonic wave to describe such a pulse is too complicated. So you can say that, OK, now
let's just consider two waves and see what we can
learn from this. And this is actually what
I am going to do now. So from Bolek's
lecture I hope that he covered the beat phenomenon. So basically, what is it? A beat phenomenon? Beat phenomenon happens
when you overlap two waves, two harmonic waves. They have pretty
close wavelengths. OK, but they're not the same. And now, if you add
two waves together, that's actually what
you are going to get. You are going to get something
which is oscillating really, really fast, which is
basically called the carrier. And also you can see that the
magnitude of the oscillation is actually changing as
a function of position, and that we call envelope. So that's essentially
the beat phenomenon, which you learned from
previous lectures. So in this example, I'm going
to add two waves together. So the first wave
is described by-- OK, is denoted by side one. It's a function of x and t,
and it has a function of form A is the amplitude. And the sine k1
x minus omega1 t. This is actually
a progressing wave propagating toward
the right-hand side of the board, the positive
direction of the x-axis in my coordinate system. And it has a wavenumber of k1
and angular frequency omega1 And I can also write
down my second wave, which I would like to
overlap with the first wave. So this is actually having
exactly the same amplitude, which is A. And it is
described by a sine function, and you have a wavenumber
k2 x minus omega2 t, angular frequency omega2. With these two equations,
we can calculate the speed of propagation for
the individual waves, right? So the first one,
I can calculate the speed of propagation
v1 would be equal to omega1 over k1. Very similarly, you
can also calculate the speed of propagation
for the second wave, which is omega2 over k2. So now what I'm going
to do is to calculate a sum of these two waves. So I have the total, which is
psi is equal to psi1 plus psi2. So what I'm going to do is
to overlap these two waves and see what will happen. And for that, I
need this formula, which is a sine A
plus sine B. And this would be equal to 2 times sine
A plus B over 2 and sine-- it would become cosine here-- cosine A minus B over 2. So if I use that formula,
basically what I'm going to get is-- we have two times
from the formula. So if you have 2A sine
k1 plus k2 over 2x minus omega1 plus omega2 over 2. So basically, the first
term is the sine function. The sine function
and the content is actually A plus B. Therefore,
you add these two together, divide it by two,
then basically this is as actually what you obtain. The second term
is a cosine term. You get a cosine here. But now you calculate
A minus B, which is this term minus
that term divided by 2. Then basically what you
get is k1 minus k2 divided by 2 times x minus omega1
minus omega2 over 2 t. OK, so now this actually-- what would happen if you add
these two waves together? Until now, everything is exact. And I would like to add
additional conditions or additional assumptions
when I discuss this solution. OK? So how about in order to
produce the beat phenomenon, I need to make the
wavelengths very, very similar between the two waves. So therefore, what
I am going to do is that I'm going
to assume k1 is very close to k2 is roughly k. And because of this, since I
have a continuous function, if k1 is really
close to k2, that means omega1 is going to be also
very close to omega2, right? So what I'm going
to get is omega1 is going to be also very
similar to omega2, and I will call it omega. So if I do this, when I
have very similar k1 and k2, what is going to happen? What is going to happen is that
k1 minus k2 will be very small. So this very small k
means larger wavelengths. Therefore, this cosine term
will become the envelope, because it's actually a
slowly variating amplitude as a function of position,
because the k is very small. K is small means lambda large. Therefore, the
amplitude is going to be having this
modulation, which is actually like the envelope, that the
oscillation of this envelope is actually controlled
by the k, okay? Let's look at the
left-hand side term. k1 plus k2 over 2 is
kind of like calculating the average of the wavenumber
of the first and second wave. So if you calculate our average,
you can be still pretty large. Therefore, you have small lambda
compared to the difference. Therefore, you see
that that actually contribute to those little
structures in this graph, and it's called carrier. Yes? AUDIENCE: [INAUDIBLE]? If k1 were a lot bigger
than k2, then [INAUDIBLE].. YEN-JIE LEE: So they
can be different. Yeah, so you are
absolutely right. So you can produce
something like a carrier even when k1 is not
equal to k2, right? Its just a average. You're right. But then on the other hand,
the difference, k1 and k2 will be also large. Therefore, it's
not as easy as what we have been doing here to
identify who is the carrier and who is the envelope. But you do get
some kind of graph, which is oscillating
really fast, but the envelope is going to
be also oscillating very fast. That is harder to see
all the structure. But you're absolutely
right, yes. Very good question. So now I have this set-up. I assume that they are
very close to each other. So now I can define
phase velocity. Finally, we define what is
actually the phase velocity. In The phase velocity-- I call it vp-- you can see that
before I already have been using
phase velocity vp for the previous discussions. In the case of
nondispersive medium, the phase velocity is
just a vp, which is the velocity in the equation. And in this case, vp will
be equal to omega over k, as we discussed before. And that's actually
the definition of this phase velocity. And I can now also define
the group velocity. The group velocity is actually
the velocity of the envelope. I can calculate the
velocity of the envelope. in the case of
phase velocity, I'm calculating the
velocity of the carrier. I'm taking a ratio
of the average, and actually the average
is so close to k and omega, therefore the phase
velocity vp would be just the speed of the
propagation of the carrier, which is actually omega over k. I call it vp. And in case of group
velocity, I call it vg. vg is describing the speed of
propagation of the envelope. Therefore, what I am getting
is omega1 minus omega2 divided by k1 minus k2. Both of them have
effect of 1 over 2, which we say is canceled. And when they are really
so close to each other, this is actually
roughly like d omega dk. Any questions so far? So we have derived two
different kinds of speed. One is actually related to
the phase velocity, which is-- one is actually called
the phase velocity. It's related to the
speed of the carrier. The other one is group velocity,
which is actually related to the speed of the envelope. So let me describe you a
few interesting examples. And let's see what we can
actually learn from this. In the first example, I'm
working on a non dispersive medium, OK? If I have a
nondispersive medium, then basically what
I'm going to get is that omega will
be proportional to k. If I plot omega versus
k, it's a straight line. Now, if I have omega-- I choose the omega of
the two, omega1, omega2, of the two waves-- to be roughly equal to omega
0, I can now evaluate the vp. The vp will be the
slope of this point on the slope of
a line connecting the 0 to that point, which
is actually the omega over k, right? So that's actually
the definition of the phase velocity. I would get this slope. The slope of this
line is actually related to the phase velocity. I can also calculate
the slope of a line cuts through this point. But as it cuts through this
curve, and in this case, I'm also going to get a
line overlapping with phase velocity, because in
this case, omega over k is a constant, which is
v. Therefore, no matter what you calculate,
if you calculate vp as a ratio of omega and a k,
where you calculate vg, which is actually the slope
of the line cutting through that point, is
you always get actually v. Therefore, what we
learned from here is that for a nondispersive
medium, vp will be equal to vg. That means both of
these two curves, both of the curve of envelope,
describing the envelope and then describing
the carrier, is going to be propagating
at the same speed. OK, any questions? So the whole thing is going to
be moving at constant speed. For that, I can now show you
some example, which I prepared, some simulation
which I prepared. So what it does is
that it really-- oh, wait a second. This is 0. OK, so this is the case when
I have a nondispersive medium. if I have a nondispersive
medium, what is going to happen is that both the carrier,
which is the speed of all those little structures,
and the envelope is going to be propagating
at the same speed. So you can see the high
is like a fixed pattern. It's propagating toward
the right-hand side. And the relative motion
between the defined structure and the envelope is actually 0. So basically you have
exactly the same pattern as a function of time. So now I'm going to move away
from the nondispersive medium. How about we discuss
what would happen if we have considered the
stiffness of the string and see what we get from there. So if I plugged omega
as a function of k, and consider alpha
to be non-zero. It's a positive value. So if I have alpha to
be a positive value, non-zero, in this case, I'm
going to get a curve like this. The slope is actually
changing and it's curving up because if you
have k large, then you would see that the ratio of
omega and k actually increase. So this is actually
the kind of curve which we would get
if I set the omega of the first and
second wave of interest in this study to be omega 0. Then basically, what you
are going to get is that-- OK, now I have this
point here on the curve. If I calculate the
phase velocity-- the phase velocity, how
do I calculate that? I can now connect 0 and
the point by a line. And I can now calculate
the slope of this line, and I can get the
phase velocity, vp. On the other hand,
I can also calculate the slope of a line
cutting through, tangential to the point of interest. And that is going to give
me the group velocity. As you can see from here,
which slope is actually larger? Anybody know? Can point it out? Group velocity's larger, right? So in this case, if I
turn on alpha greater than 0, what is going to
happen is that, since the group velocity is larger than the
phase velocity, that means, if I go back to that
picture, the envelope is going to be moving faster
than the fine structure inside the envelope. How about we take a
five-minute break from here? And then we continue the
discussion after the break. It's a good time
to take a break. Welcome back, everybody. So we will continue
the discussion of the beat phenomenon. So what we have shown you is
that, based on those curves, actually can actually
determine what will be the relative velocity-- what would be the velocity of
the carrier, which is actually denoted by vp,
and the what would be the velocity of the
envelope, which is actually denoted by our group velocity. And in this case, what
I'm actually plotting here is that, in this case,
because alpha is actually greater than 0, therefore, this
curve is actually curving up. Therefore, you have
larger group velocity compared to the phase velocity. So what you would expect
is that the envelope is going to be
actually progressing at a speed higher than
the speed of the carrier. On the other hand,
if magically I can construct some
kind of medium which can be described
in this situation, alpha smaller than 0,
what is going to happen? So if I plot a situation
with alpha smaller than 0, so now I plot omega
was a function of k. What is going to happen is
that this-- so basically, you have something which is
actually curving downward. So if I now, again, work on
some point of interest here, you can see that the slope
of the phase velocity is now actually larger than
the slope, which is actually from the line
cutting through the-- tangential to the
curve, which is actually getting you the group velocity. So in the case of
alpha's more than 0, which is some
strange medium I can which I can create
from whatever, plasma, or some really strange new
kind of material of interest. If that happens, then that
means your group velocity will be smoother than
the phase velocity. And if you look at
this point here, you can see that this curve
actually reach a maxima here. And if you actually are
operating at this point, what is going to happen? What is going to happen is
that if you calculate the group velocity, what
will be the value? It will be 0. What does that mean? That means the envelope
will not be moving a lot, but the carriers
are still moving. So at this point,
you are going to get group velocity equal to 0. And finally, the if you
actually going to a very large k value in this scenario,
alpha smaller than 0, you will see that
even you can have phase velocity, vp, positive,
because it's actually a positive slope. And that the group velocity
actually is negative. What does that mean? That means you are going to see
a situation that the carriers are progressing in the
positive direction, and the envelope is
going to be progressing in the negative
direction, probably progressing to the
left-hand side of the board. So what does that mean? That means this wave is doing
what Michael Jackson's doing. It's actually
doing the moonwalk. [LAUGHTER] So this is actually
the kind of thing which could have happened,
that it looks like and that you are doing-- going forward, because
all the carriers are moving in a
positive direction. But the body is actually going
toward negative direction. maybe I can also learn
moonwalk at some point. [LAUGHTER] OK. So let's go back to
the demonstration which I got started, and
somehow I got messed up. So let's take a look
at the demo again so let's look at all the
different situation at once. So in this case, as
we discussed before, this is actually happening in
the nondispersive situation. In this situation, you
have a straight line, nondispersed medium
actually give you always the group velocity
equal to phase velocity. So that means the
carrier and the envelope is going to be moving
in the same direction at the same speed. On the other hand,
in this case, we can actually have a situation
that the phase velocity is actually faster than
the group velocity. So what I mean is actually
the situation here. The phase velocity calculated
from a line connecting from 0 to that point is actually
having a larger slope compared to the tangential line. And you see this situation. So basically, you see
that inside the envelope all those carriers are
actually moving faster than the envelope. Now I can have a dispersive
medium where the group velocity is equal to 0. So what is going to happen
is that really the envelope is actually not moving. It's not like this. The body is not moving. So you have some carriers
inside this structure is actually moving forward. But the envelope is
actually not moving. So, finally the last situation
is really interesting. So in this situation,
this is actually having the group velocity-- the group velocity is actually
having difference sine compared to the phase velocity. So you can see that the whole
structure of the envelope is actually moving backwards. But the carrier
is actually moving in the positive direction
in this example. So this is actually what we
have learned from this beat phenomenon, and
then we have covered the idea of phase velocity
and the group velocity. So how about bound
system how do we understand when we
have a bound system? And how does that evolve
as a function of time? So if I have a system of
two walls and one string, and of course, I
give you the density for the unit length and the
string tension, and also the alpha, which is
actually telling you about the stiffness
of the system. Again, I can write
down psi xt to be the sum of all the normal
mode from one to infinity, A m sine km x plus alpha m
sine omega mt plus beta m. And then what we can
do is that we can first get the initial
conditions of this system, and those are the boundary
conditions of this system. That we actually just follow
exactly the same procedure to obtain all the
unknown coefficients that we would be able to evolve
this system as a function of time, as I have demonstrated
to you in the beginning of the lecture. So in this case, you can
have two boundary conditions. One is actually say
at x equal to 0. And the other one is
actually at x equal to L. In those boundaries,
as we actually learned before, because
the endpoints are fixed on the wall. Therefore, psi of
0 at that time, t, will be always equal to 0 for
the left-hand side boundary condition. And very similarly, as
we discussed before, psi of L t will be equal to 0 if
you look at the right-hand side of the wall-- of the system. So I don't want to repeat
this, because this is actually exactly the same calculation
which we have done before. So with these two
boundary conditions, we can actually conclude that k
m will be equal to m pi over L, and alpha m will be equal to 0. So you can actually go
back and check this out. So what I'm going to
say is that until now, what we have been doing
is identical to what we have been doing for
the nondispersive media. What I'm to say is that the
shape of the normal mode is actually set by the
boundary condition. It's determined by the
boundary condition, and it has actually,
so far, nothing to do with the
dispersion relation omega as a function of k. So in short, boundary
condition can give you the shape of the normal
mode, and that we know that the first normal
mode, second normal mode, et cetera, et
cetera, is actually going to be identical to the
case of nondispersive medium. so that's actually the first
thing which we learned. The second thing we learned
is that OK, now what we see is that once the boundary
condition is given, then the k m is
actually also given. Therefore, since I have the
dispersion relation omega as a function of
k, as shown there. Omega over k is equal to
v times square root of 1 plus alpha k squared. Therefore, once k m is
given, omega m is also given. So you can see that that's
actually where the dispersion relation come into play. The omega m will be different if
you compare the dispersive case and nondispersive case. So that is actually
what I want to say. The k m, which is the
shape of the normal mode, doesn't depend on the
dispersion relation. On the other hand, the
speed of the oscillation, the angular frequency,
omega, depends on the dispersion
relation, which is actually what we obtained from there. If I start to plot omega
m as a function of k m-- so in the case of
nondispersive medium, so what am I going to get
is actually discrete points along a straight line. This is actually k1,
k2, k3, k4, et cetera. They are actually all sitting
on a common straight line. If you look at the relative
difference between k1, k2, and k3, they are constant
according to this formula. The difference between
k1 and k2 is pi over 2. k2 and k3 is actually
also pi over 2-- pi over L. It's
always a fixed number. And since omega is
actually proportional to k. Therefore, the spacing between
omega 1, omega 2, omega 3, is also constant. In short, omega 2,
omega 3, and omega 4, et cetera is always
multiple times what you get from omega
1, according to this graph and in the case of
nondispersive medium. So what does that mean? That means OK, now if I
have a very complicated initial condition-- this is actually what I have,
an initial condition-- very complicated. I just need to wait. If this is actually
nondispersive medium, I just have to wait until p
equal to 2 pi over omega 1. Then the system would restore
to its original shape. That's actually what
I can learn from here, because omega 2, omega 3, and
any higher order normal modes, the angular frequency is
actually multiple times of what I get from omega 1. On the other hand, if
I consider a situation of dispersive medium-- you can see that now the
difference between omega m is now the constant. So what you would predict
is that it would take much, much longer for this
system to go back to the original shape compared
to nondispersive media. So that actually you
can actually see. In a real-life experiment,
I can distort this equipment in this boundless
system, and it's actually going to take forever or
impossible to come back to the original shape,
because of that dispersion. On the other hand, if
I have a really highly idealized situation, if
I have both ends bound, and I just have to wait until
t equal to 2 pi over omega 1. Then this system will go
back to the original shape. Before I end the
lecture today, I would like to discuss with
you two interesting issues. So many of you have
seen water waves, and Feynman actually
told us in his lecture that water waves are really
easily seen by everybody, but it's actually the
worst possible example. That's the bad news-- the
worst possible example because it has all the
possible complications that waves can have. That's the bad news. The good news is that you are
going to do that in your P set. [LAUGHTER] So we will be able to understand
the behavior of the water waves. So that's the good news. The second thing which I
would like to talk about is phase velocity. You can say, OK, you say that
phase velocity or harmonic waves doesn't send
information, right? And how do I actually know that? Right? So what does that mean? OK, so let's take this
horrible example of water wave. OK, so the black line
is actually the beach, and there is a water wave
from the ocean approaching the beach. And you can see
that you can have some kind of angle between
the insert of water wave and the line of the beach. What I can actually
do is that I can now measure the shape of the water
wave at the edge of the beach. And I would see that, huh,
now the phase velocity which I observe
there is actually faster than the speed of
propagation of the water wave, because of this
inserted angle, OK? I can actually make
it very, very fast. I can make the speed
actually even faster than the speed of light. Right? I can now decrease
the theta to 0. Then you will have
a phase velocity which is faster than
the speed of light. It goes to infinity. But does that mean anything? Actually, that
doesn't mean anything, because I don't really move
the water from a specific point to another point
infinitely fast. Therefore, what I want
to say is that, OK, you can do whatever you want
to make a fancy phase velocity. But that will not
help you with sending things close to the
speed of light or greater than the speed of light. So as you can see
from this example, I can easily construct
a simple example, which you see that is actually
really not sending anything from one place to the other. But you still have really,
really fast phase velocity. OK, thank you very
much, everybody, for the attention and hope
you enjoyed the lecture. And if you have any
questions, please let me know.
