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YEN-JIE LEE: So
let's get started.
00:00:25.880 --> 00:00:29.180
This is our goal for 8.03.
00:00:29.180 --> 00:00:32.299
So you can see, during
the exam number one,
00:00:32.299 --> 00:00:35.120
we have covered the
first half of the goal,
00:00:35.120 --> 00:00:37.640
and we are actually
making progress
00:00:37.640 --> 00:00:40.520
to learn about
boundary conditions
00:00:40.520 --> 00:00:42.860
in one dimensional
system and also
00:00:42.860 --> 00:00:44.930
in two dimensional system today.
00:00:44.930 --> 00:00:48.200
And we will actually talk
about phenomena related
00:00:48.200 --> 00:00:51.350
to electromagnetic
waves and optics
00:00:51.350 --> 00:00:54.800
today, which we will
be able to learn
00:00:54.800 --> 00:00:58.090
two very important
fundamental laws related
00:00:58.090 --> 00:01:05.370
to geometrical optics.
00:01:05.370 --> 00:01:08.620
OK, so that's the excitement.
00:01:08.620 --> 00:01:12.830
And then we started a discussion
of two dimensional or three
00:01:12.830 --> 00:01:15.500
dimensional wave last time.
00:01:15.500 --> 00:01:19.760
And just in case you
haven't realized that,
00:01:19.760 --> 00:01:23.780
there are two ways to
go to higher dimension.
00:01:23.780 --> 00:01:28.900
So the first way is to
increase the number of objects
00:01:28.900 --> 00:01:31.650
and place that in two
dimensional or three
00:01:31.650 --> 00:01:33.230
dimensional space.
00:01:33.230 --> 00:01:38.190
And that is the kind of things
which you will discuss today.
00:01:38.190 --> 00:01:40.340
So for example, I
can have particles
00:01:40.340 --> 00:01:44.770
arranged in two dimensions
which form membranes.
00:01:44.770 --> 00:01:47.750
And then we can also,
on the other hand,
00:01:47.750 --> 00:01:51.830
change the direction of
the electromagnetic wave,
00:01:51.830 --> 00:01:54.360
for example, as a
function of time,
00:01:54.360 --> 00:01:59.270
and that's another way to
go to a higher dimension.
00:01:59.270 --> 00:02:01.580
And today, as I
mentioned before, we
00:02:01.580 --> 00:02:05.420
are going to talk about the
first case, and on Thursday
00:02:05.420 --> 00:02:07.730
we are going to talk
about the second way
00:02:07.730 --> 00:02:11.280
to go to higher dimension, which
is related to polarization,
00:02:11.280 --> 00:02:11.820
et cetera.
00:02:14.510 --> 00:02:22.390
In general, higher order
dimensions are hopeless.
00:02:22.390 --> 00:02:24.930
They are super complicated.
00:02:24.930 --> 00:02:27.490
And, in general, we
don't really know how
00:02:27.490 --> 00:02:30.680
to solve this kind of system.
00:02:30.680 --> 00:02:34.810
Fortunately, in 8.03,
what we have been doing
00:02:34.810 --> 00:02:41.020
is focusing on a small subset of
questions of which are actually
00:02:41.020 --> 00:02:43.240
highly symmetric.
00:02:43.240 --> 00:02:45.860
Therefore, we can actually
solve it analytically.
00:02:45.860 --> 00:02:49.630
So that will be
the focus of 8.03,
00:02:49.630 --> 00:02:52.360
so that we can actually learn
some physics intuition out
00:02:52.360 --> 00:02:56.310
of this kind highly
idealized system.
00:02:56.310 --> 00:03:00.750
And the system which we
are going to focus on today
00:03:00.750 --> 00:03:02.200
is shown here.
00:03:02.200 --> 00:03:04.200
It's a two dimensional
system, which
00:03:04.200 --> 00:03:12.190
you have array of masses
placing the x and y, x, y plan.
00:03:12.190 --> 00:03:17.860
And that is the system we
are going to solve today.
00:03:17.860 --> 00:03:21.720
And we will learn a lot of
interesting phenomena coming
00:03:21.720 --> 00:03:24.870
from the solution of
this kind of system.
00:03:24.870 --> 00:03:28.020
Before we start a discussion
of two dimensional system,
00:03:28.020 --> 00:03:30.780
I would like to remind you
of what we have already
00:03:30.780 --> 00:03:32.970
learned from lecture eight.
00:03:32.970 --> 00:03:37.950
So that was about a system which
consists of infinite number
00:03:37.950 --> 00:03:42.330
of mass and the infinite number
of strings, and each string
00:03:42.330 --> 00:03:46.020
have string tension T.
And all the mass, when
00:03:46.020 --> 00:03:49.920
they are in
equilibrium position,
00:03:49.920 --> 00:03:53.690
the distance between all
those mass in the x direction
00:03:53.690 --> 00:03:55.320
is a, OK?
00:03:55.320 --> 00:04:00.600
So we have solved this system
before with space translation
00:04:00.600 --> 00:04:02.210
symmetry.
00:04:02.210 --> 00:04:06.900
And this is just a reminder that
the dispersion relation, which
00:04:06.900 --> 00:04:11.160
we got a lot time, omega,
as a function of k,
00:04:11.160 --> 00:04:15.510
is t over ma sine Ka over 2.
00:04:15.510 --> 00:04:18.000
So that was just a
reminder of what we have
00:04:18.000 --> 00:04:20.040
learned from lecture eight.
00:04:20.040 --> 00:04:24.750
So by now you should
realize that, OK,
00:04:24.750 --> 00:04:28.860
dispersion relation is unusual.
00:04:28.860 --> 00:04:31.170
This is actually
telling you that this
00:04:31.170 --> 00:04:34.320
is a dispersive media, right?
00:04:34.320 --> 00:04:37.710
Because if you calculate
the ratio of omega and k,
00:04:37.710 --> 00:04:40.260
you'll see that this is
actually not a constant.
00:04:40.260 --> 00:04:43.150
So after all the discussion
from previous lecture,
00:04:43.150 --> 00:04:46.800
you should be able to
immediately realize that.
00:04:46.800 --> 00:04:52.190
And any wave propagating
on this kind of system,
00:04:52.190 --> 00:04:56.220
there will be a dispersion
phenomena happening
00:04:56.220 --> 00:04:58.350
in this kind of system.
00:04:58.350 --> 00:04:59.510
OK?
00:04:59.510 --> 00:05:03.150
And also from the
previous lecture,
00:05:03.150 --> 00:05:07.220
we'll have learned that
the eigenvectors based
00:05:07.220 --> 00:05:09.620
on space translation
symmetry, it's
00:05:09.620 --> 00:05:17.150
exponential of ikx, where x
is defined as j times a, where
00:05:17.150 --> 00:05:21.050
it's a is a label to
tell you what which mass
00:05:21.050 --> 00:05:23.870
I was talking about.
00:05:23.870 --> 00:05:28.190
Now today, we are
going to extend this
00:05:28.190 --> 00:05:31.160
to a two dimensional system.
00:05:31.160 --> 00:05:34.460
So instead of a one
dimensional system
00:05:34.460 --> 00:05:37.980
we have a two dimensional array.
00:05:37.980 --> 00:05:42.950
So all the little mass
all have mass equal to n,
00:05:42.950 --> 00:05:46.370
and they are placing xy plan.
00:05:46.370 --> 00:05:50.010
The coordinate system,
which I defined is here.
00:05:50.010 --> 00:05:53.840
x is horizontal, and
the y is vertical,
00:05:53.840 --> 00:05:57.090
and the z is actually
pointing to you.
00:05:57.090 --> 00:05:59.690
And all those
little mass can only
00:05:59.690 --> 00:06:05.010
oscillate toward you
or going away from you,
00:06:05.010 --> 00:06:07.070
so in the z direction.
00:06:07.070 --> 00:06:12.470
It can only oscillate up
and down in the z direction.
00:06:12.470 --> 00:06:16.250
And in this system we have
the length scale, which
00:06:16.250 --> 00:06:21.310
is the horizontal distance
between mass, is called aH.
00:06:21.310 --> 00:06:24.350
And in the vertical
direction, the scale
00:06:24.350 --> 00:06:29.220
of the distance
between mass is av.
00:06:29.220 --> 00:06:31.500
Also, we have string tension--
00:06:31.500 --> 00:06:33.380
two different kinds
of string tension
00:06:33.380 --> 00:06:36.080
for the vertical and
horizontal direction.
00:06:36.080 --> 00:06:39.230
The vertical direction,
you have string tension Tv,
00:06:39.230 --> 00:06:44.040
and in the horizontal direction
you have string tension Th.
00:06:44.040 --> 00:06:44.810
OK
00:06:44.810 --> 00:06:48.680
So how do we actually describe
this kind of system, right?
00:06:48.680 --> 00:06:52.520
The first thing,
as we did before,
00:06:52.520 --> 00:06:57.770
is to label those
little mass by my label.
00:06:57.770 --> 00:07:01.770
And my label is
called Jx and Jy,
00:07:01.770 --> 00:07:09.960
which tells you which mass I was
talking about in this system.
00:07:09.960 --> 00:07:13.580
Once I have defined
that, the labels,
00:07:13.580 --> 00:07:19.220
I will be able to write the
position of all those mass,
00:07:19.220 --> 00:07:22.490
the x direction position
and y direction position,
00:07:22.490 --> 00:07:26.540
in terms of J and the
A. So for example,
00:07:26.540 --> 00:07:32.510
x position of 1 over the mass
will be written as Jx times ah.
00:07:32.510 --> 00:07:35.630
And y position of
a specific mass,
00:07:35.630 --> 00:07:39.820
you can write it down in
terms of a Jy times av.
00:07:39.820 --> 00:07:44.420
So all those things should
be pretty straight forward.
00:07:44.420 --> 00:07:47.000
The interesting
part is that, as we
00:07:47.000 --> 00:07:50.720
identified in the last
lecture, this system
00:07:50.720 --> 00:07:52.790
is highly symmetric.
00:07:52.790 --> 00:07:56.220
It has space translation
symmetry, right?
00:07:56.220 --> 00:08:00.200
Therefore, we can actually
immediately figure out
00:08:00.200 --> 00:08:04.430
what will be the
eigenvectors for this system.
00:08:04.430 --> 00:08:08.150
So the eigenvector--
very similar to what
00:08:08.150 --> 00:08:10.550
has been discussed here,
where you have a one
00:08:10.550 --> 00:08:14.480
dimensional space
translation symmetric system.
00:08:14.480 --> 00:08:18.170
Exponential of ikx
was the eigenvector.
00:08:18.170 --> 00:08:22.100
Now you have eigenvector
which is in two dimension,
00:08:22.100 --> 00:08:24.410
because you would
like to describe
00:08:24.410 --> 00:08:27.690
not only the x direction
but also y direction.
00:08:27.690 --> 00:08:31.820
And the eigenvector have
exactly the same functional form
00:08:31.820 --> 00:08:34.270
because of space
translational symmetry,
00:08:34.270 --> 00:08:40.460
and it is like exponentional
of ixk times x.
00:08:40.460 --> 00:08:43.250
Multiply that by another
exponential function--
00:08:43.250 --> 00:08:47.470
exponential iky times y.
00:08:47.470 --> 00:08:54.770
So I think, until now,
nothing should surprise you
00:08:54.770 --> 00:08:58.130
because this is what we
have learned from the one
00:08:58.130 --> 00:09:02.800
dimensional system analysis.
00:09:02.800 --> 00:09:05.180
Based on what we
have learned before,
00:09:05.180 --> 00:09:07.750
we can also
immediately write down
00:09:07.750 --> 00:09:09.820
what would be the
dispersion relation.
00:09:09.820 --> 00:09:14.570
Since we are always considering
very small vibration,
00:09:14.570 --> 00:09:18.940
and this formula
is still applies,
00:09:18.940 --> 00:09:23.080
therefore you can actually write
down the dispersion relation--
00:09:23.080 --> 00:09:42.260
omega squared k will be equal
to 4 Th over Mah sine square Kx
00:09:42.260 --> 00:09:56.940
Ah, divided by 2 plus 4 Tv
divided by Mav sine square Ky
00:09:56.940 --> 00:10:02.040
Av divided by 2.
00:10:02.040 --> 00:10:08.160
So this is actually
pretty straightforward.
00:10:08.160 --> 00:10:18.440
And you can see that omega is a
function of both Kx and the Ky.
00:10:18.440 --> 00:10:22.580
From the eigenvector
we can also write down
00:10:22.580 --> 00:10:26.930
what would be the
possible Psi xy.
00:10:26.930 --> 00:10:29.490
Now the Psi is actually
the displacement
00:10:29.490 --> 00:10:33.560
in the z direction, with respect
to the equilibrium position.
00:10:33.560 --> 00:10:36.980
And that is actually
proportional
00:10:36.980 --> 00:10:38.810
to the eigenvector.
00:10:38.810 --> 00:10:45.635
So basically it's going to
be a exponential ikx times x
00:10:45.635 --> 00:10:50.690
exponential iky times y.
00:10:50.690 --> 00:10:54.830
And of course I can write these
two terms together, right?
00:10:54.830 --> 00:11:00.140
So basically what I would
get is a exponential i,
00:11:00.140 --> 00:11:05.720
k is a vector times
r, which is a vector.
00:11:05.720 --> 00:11:10.700
So basically k contains
two components, Kx and Ky,
00:11:10.700 --> 00:11:14.110
and r have also two
components, which is x and y.
00:11:17.820 --> 00:11:21.580
Again, we see that this is
actually a non dispersive
00:11:21.580 --> 00:11:23.700
medium.
00:11:23.700 --> 00:11:28.440
And what we are going to do
is to make linear combination
00:11:28.440 --> 00:11:32.190
of all those eigenvectors
and figure out
00:11:32.190 --> 00:11:36.900
what would be the behavior
when this system is oscillating
00:11:36.900 --> 00:11:42.840
at a specific frequency
omega, and that is actually
00:11:42.840 --> 00:11:47.160
the corresponding normal mode
at the angular frequency omega.
00:11:47.160 --> 00:11:51.000
So that is actually pretty
similar to what we have done
00:11:51.000 --> 00:11:52.980
for one dimensional system.
00:11:56.940 --> 00:11:59.400
So this is a two
dimensional system.
00:11:59.400 --> 00:12:05.310
Just a reminder about one
dimensional system for a while.
00:12:05.310 --> 00:12:09.810
So there are two
eigenvectors which
00:12:09.810 --> 00:12:12.540
have identical omega, right?
00:12:12.540 --> 00:12:16.860
So the first one
is exponential ikx,
00:12:16.860 --> 00:12:20.550
and the second one is
exponential of minus ikx.
00:12:23.190 --> 00:12:25.980
What we have done
before is to do
00:12:25.980 --> 00:12:29.970
a linear combination of the two
exponential functions, right?
00:12:29.970 --> 00:12:32.400
So what we can do is that--
00:12:32.400 --> 00:12:36.540
OK, now I can create cosine Kx.
00:12:36.540 --> 00:12:41.510
This is actually 1
over 2 exponential ikx
00:12:41.510 --> 00:12:43.500
plus exponential minus ikx.
00:12:46.560 --> 00:12:49.440
Or I can also create sine Kx.
00:12:49.440 --> 00:12:54.900
And this is actually 1
over 2i, exponential ikx
00:12:54.900 --> 00:13:01.090
minus exponential minus ikx,
for this one dimensional system.
00:13:01.090 --> 00:13:06.180
So that's how we figure
out when the system is
00:13:06.180 --> 00:13:09.890
doing one of the normal mode.
00:13:09.890 --> 00:13:13.780
The shape of the system is
like a cosine or a sine.
00:13:13.780 --> 00:13:17.490
Or in general, you can
add these two together,
00:13:17.490 --> 00:13:19.470
and in general it
can be something
00:13:19.470 --> 00:13:24.440
like cosine Kx plus
5, but 5 is actually
00:13:24.440 --> 00:13:30.630
some phase angle, which
you can figure out
00:13:30.630 --> 00:13:32.880
by boundary condition.
00:13:32.880 --> 00:13:36.250
But before we introduce
any boundary condition,
00:13:36.250 --> 00:13:40.000
all the k values, all the
five values are allowed.
00:13:40.000 --> 00:13:44.040
Just a reminder about what
we have learned before.
00:13:44.040 --> 00:13:46.630
So the situation
is pretty simple.
00:13:46.630 --> 00:13:49.350
You have just plus
and minus k, and then
00:13:49.350 --> 00:13:51.720
you make linear
combination of these two,
00:13:51.720 --> 00:13:57.350
then you know what will be
the shape of the normal mode.
00:13:57.350 --> 00:14:01.190
On the other hand,
we are now talking
00:14:01.190 --> 00:14:04.430
about two dimensional case.
00:14:04.430 --> 00:14:08.330
So let's take a look at
this dispersion relation.
00:14:08.330 --> 00:14:11.540
The dispersion relation
we will have here, omega
00:14:11.540 --> 00:14:15.890
is a function of Kx, is
a function of Ky as well.
00:14:15.890 --> 00:14:18.360
OK, what does that mean?
00:14:18.360 --> 00:14:25.850
That means I can have
multiple choice of Kx and Ky,
00:14:25.850 --> 00:14:30.460
which they all produce
the same omega value.
00:14:33.000 --> 00:14:35.490
So it's not as
simple as this one
00:14:35.490 --> 00:14:38.070
any more, as you can see, right?
00:14:38.070 --> 00:14:43.920
Because when I slightly increase
Kx, what I could do is--
00:14:43.920 --> 00:14:49.530
OK, I can slightly reduce Ky
to compensate the difference.
00:14:49.530 --> 00:14:52.840
Therefore, I can
still keep omega,
00:14:52.840 --> 00:14:57.640
which is the angular frequency
of the oscillation the same.
00:14:57.640 --> 00:15:01.260
OK, so that can be seen
from this demonstration
00:15:01.260 --> 00:15:02.580
on the slide.
00:15:02.580 --> 00:15:07.290
You can see that this is
actually one example dispersion
00:15:07.290 --> 00:15:08.980
relation.
00:15:08.980 --> 00:15:12.150
This is actually the formula
which we have on the board.
00:15:12.150 --> 00:15:16.320
And what about if I set all
those parameters and get,
00:15:16.320 --> 00:15:21.030
example, omega squared
equal to 5 sine squared Kx,
00:15:21.030 --> 00:15:26.160
and the plus 5 sine squared Ky?
00:15:26.160 --> 00:15:28.960
OK, so what will happen?
00:15:28.960 --> 00:15:35.190
If I set my aht and the m value,
so that I have this example.
00:15:35.190 --> 00:15:38.520
What will happen if I have
that dispersion relation?
00:15:38.520 --> 00:15:43.260
So if I go ahead, and then
plot allowed Kx and Ky
00:15:43.260 --> 00:15:49.090
value, which gives 1, you see a
very beautiful pattern, right,
00:15:49.090 --> 00:15:49.960
on this.
00:15:49.960 --> 00:15:56.040
So you can see that ha, all
those things on the circle
00:15:56.040 --> 00:16:00.180
can produce angular
frequency omega equal to 1.
00:16:00.180 --> 00:16:02.715
And 1 will be their--
00:16:02.715 --> 00:16:04.970
1 will be the--
normal mode will be
00:16:04.970 --> 00:16:09.570
all possible linear combination
of all those possible Kx, Ky
00:16:09.570 --> 00:16:10.260
pairs.
00:16:10.260 --> 00:16:11.330
You have a question?
00:16:11.330 --> 00:16:14.830
AUDIENCE: [INAUDIBLE]
00:16:23.460 --> 00:16:25.540
YEN-JIE LEE: In
general, I think this--
00:16:25.540 --> 00:16:27.120
you mean a circular shape?
00:16:27.120 --> 00:16:29.610
AUDIENCE: [INAUDIBLE]
00:16:29.610 --> 00:16:31.470
YEN-JIE LEE: I think,
in general, yes,
00:16:31.470 --> 00:16:34.480
you do have determinacy
because you can--
00:16:34.480 --> 00:16:39.450
but the shape will not
be circular, for example.
00:16:39.450 --> 00:16:41.970
OK, so it can be
a general function
00:16:41.970 --> 00:16:47.640
which is like the formula
above, but this argument still
00:16:47.640 --> 00:16:48.330
applies.
00:16:48.330 --> 00:16:51.390
So if you have some
intermediate omega value,
00:16:51.390 --> 00:16:56.010
you can always slightly increase
Kx and slightly decrease Ky,
00:16:56.010 --> 00:17:00.030
and that will still satisfy
the same omega value.
00:17:00.030 --> 00:17:02.085
Therefore, all the
normal modes, we
00:17:02.085 --> 00:17:05.339
set specific omega will be
a linear combination of all
00:17:05.339 --> 00:17:07.990
those possible normal mode.
00:17:07.990 --> 00:17:11.880
All those possible Kx
and Ky pairs if you
00:17:11.880 --> 00:17:15.720
have an infinitely long system.
00:17:15.720 --> 00:17:19.710
And the law also applies
to the other example, when
00:17:19.710 --> 00:17:22.290
I have omega equal
to 5, then you have
00:17:22.290 --> 00:17:23.579
slightly different behavior.
00:17:23.579 --> 00:17:25.710
But the take home
message is that there
00:17:25.710 --> 00:17:28.109
are many, many
pairs of Kx and Ky,
00:17:28.109 --> 00:17:34.890
which can create the
same amount of omega.
00:17:34.890 --> 00:17:37.670
So that makes things
pretty complicated.
00:17:37.670 --> 00:17:42.050
Potentially, we can
always still try
00:17:42.050 --> 00:17:44.790
to understand this
by investigating
00:17:44.790 --> 00:17:48.910
all the possible k
pairs of Kx and Ky.
00:17:48.910 --> 00:17:50.140
On the other hand--
00:17:53.080 --> 00:17:55.350
this is what we have
discussed before.
00:17:55.350 --> 00:17:58.840
Before you introduce
boundary condition
00:17:58.840 --> 00:18:02.290
for the one dimensional
system, there
00:18:02.290 --> 00:18:05.720
are infinite number of
possible k value, right?
00:18:05.720 --> 00:18:08.530
All the possible k
values are allowed.
00:18:08.530 --> 00:18:13.550
But after you add boundary
condition-- for example,
00:18:13.550 --> 00:18:20.045
I add walls around this system,
so that I basically have
00:18:20.045 --> 00:18:22.480
a fixed boundary condition.
00:18:22.480 --> 00:18:24.730
So basically, the
boundary condition
00:18:24.730 --> 00:18:30.370
is that the amplitude at x
equal to 0, y equal to 0, or x
00:18:30.370 --> 00:18:38.550
equal to 5ah or y equal to 4av.
00:18:38.550 --> 00:18:43.960
At the boundary, the amplitude
has to be equal to 0,
00:18:43.960 --> 00:18:46.960
because it is
attached to a wall.
00:18:46.960 --> 00:18:53.510
OK, when this happens
this means that we
00:18:53.510 --> 00:18:59.750
will have have four
wall which will
00:18:59.750 --> 00:19:02.430
have a corresponding
boundary condition.
00:19:02.430 --> 00:19:06.700
So that means I have to
satisfy this four boundary
00:19:06.700 --> 00:19:13.580
consideration of side 0,y
evaluated at any time will be
00:19:13.580 --> 00:19:25.790
equal to Psi Lh, y, t,
will be equal to Psi x,0,t,
00:19:25.790 --> 00:19:29.555
will be equal to Psi x,Lv,t.
00:19:32.750 --> 00:19:36.590
And this is all equal to 0.
00:19:36.590 --> 00:19:40.610
So those are not very
difficult to understand.
00:19:40.610 --> 00:19:47.170
Those are just the four
walls around the system.
00:19:47.170 --> 00:19:51.350
Once you have all
those conditions--
00:19:51.350 --> 00:19:57.530
and of course I define Lh
will be equal to 5 times Ah,
00:19:57.530 --> 00:20:05.150
because there are 5 strings
between the two walls
00:20:05.150 --> 00:20:07.160
in the horizontal direction.
00:20:07.160 --> 00:20:10.110
And of course I also
have defined here,
00:20:10.110 --> 00:20:12.610
Lv will be 4 times equal to av.
00:20:16.810 --> 00:20:20.550
So once I have all those four
boundary conditions in place
00:20:20.550 --> 00:20:25.890
that means I cannot arbitrarily
choose k value and the fact,
00:20:25.890 --> 00:20:26.640
right?
00:20:26.640 --> 00:20:31.050
Otherwise, I will not be able
to satisfy these four boundary
00:20:31.050 --> 00:20:33.870
conditions.
00:20:33.870 --> 00:20:39.300
So now we actually will
be able to figure out
00:20:39.300 --> 00:20:44.490
that there will be only
four modes in this two
00:20:44.490 --> 00:20:48.890
dimensional problem, which
will give the same omega.
00:20:48.890 --> 00:20:52.140
What are the four
possible nodes--
00:20:52.140 --> 00:20:56.250
what are the four
possible eigenvectors?
00:20:56.250 --> 00:21:02.520
Those are a exponential
plus or minus ikx
00:21:02.520 --> 00:21:15.150
times x, exponential plus minus
iky times y, where the Kx--
00:21:15.150 --> 00:21:16.930
because of the
boundary condition,
00:21:16.930 --> 00:21:20.730
which we have solved in the
one dimensional system--
00:21:20.730 --> 00:21:26.690
Kx will be equal to
Nx pi divided by Lh--
00:21:34.000 --> 00:21:40.720
in order to match the boundary
condition, add x equal to 0
00:21:40.720 --> 00:21:45.130
and x equal to Lh.
00:21:45.130 --> 00:21:53.950
And Ky will be equal to
y times pi divided by Lv.
00:21:53.950 --> 00:21:57.490
That's actually
designed to match
00:21:57.490 --> 00:22:02.410
the boundary condition
at y equal to 0
00:22:02.410 --> 00:22:05.990
and then y equal to Lv.
00:22:05.990 --> 00:22:09.040
So you can see that,
like when we've
00:22:09.040 --> 00:22:14.090
seen before with one dimensional
system, after you introduced
00:22:14.090 --> 00:22:18.460
the boundary condition it's not
an infinity long system anymore
00:22:18.460 --> 00:22:24.400
that allowed k value, which
is the length number in the x
00:22:24.400 --> 00:22:25.830
and y direction.
00:22:25.830 --> 00:22:31.150
For example, in this case,
it's also become limited,
00:22:31.150 --> 00:22:37.960
and only a limited number of
possible values are allowed.
00:22:37.960 --> 00:22:45.970
In this case, Nx is allowed
to be equal to 1, 2 until 4,
00:22:45.970 --> 00:22:56.496
and Ny is equal 1,2,3 in this
system we are talking about.
00:22:56.496 --> 00:22:57.370
Any questions so far?
00:23:00.021 --> 00:23:00.520
Yep?
00:23:00.520 --> 00:23:05.018
AUDIENCE: I think you
mentioned that Kx and Ky are
00:23:05.018 --> 00:23:06.976
directly related rather
than inversely related,
00:23:06.976 --> 00:23:08.928
but I'm sort of confused
as to why that is.
00:23:08.928 --> 00:23:11.368
Because if you want to
maintain the frequency,
00:23:11.368 --> 00:23:13.320
it increases the wave
numer and [INAUDIBLE]..
00:23:17.430 --> 00:23:19.960
YEN-JIE LEE: Yeah,
so I was talking
00:23:19.960 --> 00:23:26.240
about when I choose Kx and Ky
in the infinitely long system.
00:23:26.240 --> 00:23:29.270
OK, all of the possible
values of Kx and Ky
00:23:29.270 --> 00:23:32.830
are allowed because I have
a infinitely long system
00:23:32.830 --> 00:23:34.730
with no boundary condition.
00:23:34.730 --> 00:23:40.450
And in that case, going back
to this dispersion relation,
00:23:40.450 --> 00:23:42.870
I have the freedom to--
00:23:42.870 --> 00:23:46.150
OK, so when I increase
a little bit, Kx,
00:23:46.150 --> 00:23:49.720
I can always decrease
a little bit, the Ky.
00:23:49.720 --> 00:23:54.520
OK, so the question is why
that's not the case, right,
00:23:54.520 --> 00:23:56.130
for the discrete case.
00:23:56.130 --> 00:24:01.690
As you can see from here, after
we introduced the boundary
00:24:01.690 --> 00:24:04.590
condition, the four
boundary conditions
00:24:04.590 --> 00:24:11.620
especially describe the
boundary of the four walls.
00:24:11.620 --> 00:24:15.040
And what is going to happen
is that you will also see that
00:24:15.040 --> 00:24:19.810
the allowed Kx value
is becoming limited,
00:24:19.810 --> 00:24:23.830
because you cannot arbitrarily
choose with lengths, right,
00:24:23.830 --> 00:24:27.340
if you choose a side along the
wavelengths like what we have
00:24:27.340 --> 00:24:29.840
been trying to do for the
infinitely long system--
00:24:29.840 --> 00:24:33.460
not that it matched
the boundary condition.
00:24:33.460 --> 00:24:36.850
Therefore, you don't have
this degree of freedom
00:24:36.850 --> 00:24:41.880
to choose slightly
higher or slightly lower
00:24:41.880 --> 00:24:46.380
Ky when I change a Kx.
00:24:46.380 --> 00:24:49.900
So you can see that the
allowed value are discrete.
00:24:49.900 --> 00:24:53.380
Therefore, the number
of possible combinations
00:24:53.380 --> 00:24:55.870
of Kx and Ky is also limited.
00:24:55.870 --> 00:24:59.650
And in this case, it's
actually very likely
00:24:59.650 --> 00:25:02.920
to be limited to be only
four pairs, which is actually
00:25:02.920 --> 00:25:05.330
plus, minus Kx and
the plus, minus Ky.
00:25:07.730 --> 00:25:08.230
All right.
00:25:11.420 --> 00:25:12.880
Thank you for the question.
00:25:12.880 --> 00:25:15.800
OK, then once I
have those I can do
00:25:15.800 --> 00:25:21.560
a linear combination of these
four possible eigenvectors.
00:25:21.560 --> 00:25:23.360
And also, at the
same time, I will
00:25:23.360 --> 00:25:26.870
try to match the
boundary condition.
00:25:26.870 --> 00:25:30.250
So if I jump forward,
basically what you can conclude
00:25:30.250 --> 00:25:38.570
is that Psi Nx and y, so
that's with an Nx value y
00:25:38.570 --> 00:25:43.910
value chosen for the
determination of Kx and Ky.
00:25:43.910 --> 00:25:50.020
And is this actually a function
of x and y and of course also
00:25:50.020 --> 00:25:52.560
time, when I also
make it oscillate
00:25:52.560 --> 00:25:54.060
as a function of time.
00:25:54.060 --> 00:25:57.920
This will be equal to
some arbitrary constant, A
00:25:57.920 --> 00:26:08.710
of amplitude Nx,
Ny sine Nx pi x,
00:26:08.710 --> 00:26:22.470
divided by Lh sine Ny
times y divided by Lv.
00:26:22.470 --> 00:26:27.240
And of course, you can see that
this is actually sine, right?
00:26:27.240 --> 00:26:29.670
It's actually, the same as
what we have done for the one
00:26:29.670 --> 00:26:31.030
dimensional system, right?
00:26:31.030 --> 00:26:34.185
So if you have two boundary
conditions that said,
00:26:34.185 --> 00:26:38.200
look, the beginning
and the end, therefor,
00:26:38.200 --> 00:26:41.890
the corresponding normal mode
is always a sine function.
00:26:41.890 --> 00:26:45.360
So that's what we have
learned from the one
00:26:45.360 --> 00:26:46.410
dimensional system.
00:26:46.410 --> 00:26:51.390
And this is also the case for
the two dimensional system.
00:26:51.390 --> 00:26:57.030
And of course, don't
forget this wave function
00:26:57.030 --> 00:26:59.730
is changing as a
function of time
00:26:59.730 --> 00:27:03.420
oscillating up and
down harmonically.
00:27:03.420 --> 00:27:14.200
Therefore, you have sine omega
Nx, Ny times T plus Theta,
00:27:14.200 --> 00:27:18.520
which is a phase
to be determined
00:27:18.520 --> 00:27:21.610
by initial conditions.
00:27:21.610 --> 00:27:27.730
And you can see that the
whole equation, a sine sine
00:27:27.730 --> 00:27:31.620
is multiplied by a sine
omega T plus 5 because
00:27:31.620 --> 00:27:33.340
of beta function, right?
00:27:33.340 --> 00:27:36.960
So that means the
shape is actually
00:27:36.960 --> 00:27:39.820
going up and down harmonically.
00:27:39.820 --> 00:27:43.240
So the shape is fixed,
which is sine times sine,
00:27:43.240 --> 00:27:45.960
and the whole thing
is oscillating
00:27:45.960 --> 00:27:48.790
at the same frequency
at the same phase, which
00:27:48.790 --> 00:27:51.410
is the definition of
normal mode, right?
00:27:51.410 --> 00:27:52.570
Just a reminder.
00:27:52.570 --> 00:27:56.740
And how do we actually imagine
what is actually happening?
00:27:56.740 --> 00:27:59.980
That brings me to
the demonstration,
00:27:59.980 --> 00:28:05.980
so we can really visualize
how this kind of system
00:28:05.980 --> 00:28:11.470
will look like by a
little simulation.
00:28:11.470 --> 00:28:21.340
So, suppose I choose
Nx and Ny equal to 1
00:28:21.340 --> 00:28:23.250
and see what will happen.
00:28:23.250 --> 00:28:28.550
This is the kind of oscillation
you will expect, right?
00:28:28.550 --> 00:28:33.820
So if you choose Nx equal
to 1, Ny equal to 1,
00:28:33.820 --> 00:28:36.070
then this is a system.
00:28:36.070 --> 00:28:40.360
Basically you have sine
function with no node
00:28:40.360 --> 00:28:43.690
in x and y direction.
00:28:43.690 --> 00:28:49.840
Therefore, if you do
get this simulation,
00:28:49.840 --> 00:28:55.830
you can see that there will
be no node in the x,y plane,
00:28:55.830 --> 00:29:02.770
and all those particles
are either going toward you
00:29:02.770 --> 00:29:05.620
or going away from you.
00:29:05.620 --> 00:29:13.190
They only oscillate in the z
direction in this simulation.
00:29:13.190 --> 00:29:16.120
And also, you can see that now
I can increase, for example--
00:29:19.930 --> 00:29:24.980
I can increase the Kx by
setting Nx to be 2 and see
00:29:24.980 --> 00:29:26.190
what will happen.
00:29:26.190 --> 00:29:30.200
So what is going to happen
is that if I have higher
00:29:30.200 --> 00:29:36.110
Kx in the x direction-- so the
next possible normal mode is
00:29:36.110 --> 00:29:43.510
that you have a full sine
wave in the x direction, then
00:29:43.510 --> 00:29:46.610
you are going to
see two components
00:29:46.610 --> 00:29:49.730
in this demonstration.
00:29:49.730 --> 00:29:54.980
And one part of the system
is actually moving toward you
00:29:54.980 --> 00:29:58.780
while the other half
part of the system
00:29:58.780 --> 00:30:00.680
is actually moving
away from you.
00:30:00.680 --> 00:30:05.290
And you can actually see
the node, or nodal line
00:30:05.290 --> 00:30:06.890
in this case, because
we are talking
00:30:06.890 --> 00:30:10.700
about a two dimensional
system in the middle
00:30:10.700 --> 00:30:13.730
of the distribution.
00:30:13.730 --> 00:30:16.730
Of course, we can
always go crazy, right?
00:30:16.730 --> 00:30:20.640
I can set this to a
really high value.
00:30:20.640 --> 00:30:22.310
So in this case,
the highest value
00:30:22.310 --> 00:30:27.210
I can set these is 3 and
4, and see what happens.
00:30:27.210 --> 00:30:30.290
And this is actually
a beautiful shape
00:30:30.290 --> 00:30:35.060
which is actually complicated
but understandable, as you
00:30:35.060 --> 00:30:39.380
see in this demonstration.
00:30:39.380 --> 00:30:43.760
And all those little
particles in this system
00:30:43.760 --> 00:30:48.650
are oscillating up and down
at the same angular frequency
00:30:48.650 --> 00:30:50.540
and also at the same phase.
00:30:53.450 --> 00:30:56.600
Any questions?
00:30:56.600 --> 00:31:02.100
OK, so now we have done
the discrete case, right?
00:31:02.100 --> 00:31:06.910
And of course we can also
go to the continuous case.
00:31:06.910 --> 00:31:10.800
So if we go to a
continuous limit,
00:31:10.800 --> 00:31:17.590
now I can assume that
there is a symmetry
00:31:17.590 --> 00:31:22.720
between a horizontal direction
and the vertical direction.
00:31:22.720 --> 00:31:29.460
I assume that Th is equal
to Tv is equal to T.
00:31:29.460 --> 00:31:33.690
And also I assume that the
length scale in the x direction
00:31:33.690 --> 00:31:37.030
and the y directions is
equal, and the length scale
00:31:37.030 --> 00:31:43.450
is A. In order to make the
whole system continuous,
00:31:43.450 --> 00:31:47.740
I need to increase the number
of objects in the system,
00:31:47.740 --> 00:31:52.570
and at the same time I also
need to decrease the distance
00:31:52.570 --> 00:31:54.860
between all those objects.
00:31:54.860 --> 00:31:56.970
So therefore, I need to have--
00:31:56.970 --> 00:32:00.790
this length scale goes to 0.
00:32:00.790 --> 00:32:04.690
And what is going to
happen is that if I rewrite
00:32:04.690 --> 00:32:11.050
my omega, which is a dispersion
relation, what I am going
00:32:11.050 --> 00:32:19.360
to get is 4T divided by
Na Kx squared, A squared,
00:32:19.360 --> 00:32:29.060
divided by 4, plus 4T over Na,
Ky squared a squared over 4.
00:32:32.520 --> 00:32:37.290
This is issue
because I am taking--
00:32:37.290 --> 00:32:41.880
Ah and V need to be equal
to-- and also having
00:32:41.880 --> 00:32:44.290
to be a very, very small value.
00:32:44.290 --> 00:32:50.300
Therefore, sine theta
is roughly theta, right?
00:32:50.300 --> 00:32:54.860
Therefore, I can immediately
write down this expression.
00:32:54.860 --> 00:33:03.080
And this will be equal to
Ta divided by N, Kx squared
00:33:03.080 --> 00:33:04.100
plus Ky squared.
00:33:07.780 --> 00:33:11.590
So we are facing exactly
the same situation.
00:33:11.590 --> 00:33:17.200
When I decrease A, I am
going to add more objects
00:33:17.200 --> 00:33:19.390
into the system,
but I don't want
00:33:19.390 --> 00:33:22.750
to have an infinitely
large mass.
00:33:22.750 --> 00:33:25.300
Therefore, I also
need to ensure the fix
00:33:25.300 --> 00:33:32.980
the ratio of m and a, so
that when I actually increase
00:33:32.980 --> 00:33:37.510
the number of objects, I don't
actually make the total mass
00:33:37.510 --> 00:33:40.330
go to infinity.
00:33:40.330 --> 00:33:45.770
So what I could do is I can
define Rho S is actually
00:33:45.770 --> 00:33:49.120
the surface mass density.
00:33:49.120 --> 00:33:51.460
So the surface mass
density is defined
00:33:51.460 --> 00:33:55.360
as m divided by a squared.
00:33:55.360 --> 00:33:59.770
And I can also define
a surface tension.
00:33:59.770 --> 00:34:04.060
Surface tension Ts will
be equal to T over a.
00:34:06.830 --> 00:34:12.050
And in this case, basically,
I will be able to control,
00:34:12.050 --> 00:34:16.159
so that when I increase
the number of objects,
00:34:16.159 --> 00:34:18.560
mass doesn't go
to infinity, and I
00:34:18.560 --> 00:34:24.560
have constant surface tension
and constant surface mass
00:34:24.560 --> 00:34:26.239
density.
00:34:26.239 --> 00:34:29.449
If I have defined
this to quantity
00:34:29.449 --> 00:34:38.900
then this will become Ts
divided by Rho S Kx squared
00:34:38.900 --> 00:34:43.010
plus Ky squared,
and this will be
00:34:43.010 --> 00:34:51.830
equal to Ts divided by
Rho S, k vector squared.
00:34:51.830 --> 00:34:57.230
And this k vector is a
two dimensional vector.
00:35:01.280 --> 00:35:06.020
So we are actually almost
there to make it continuous.
00:35:06.020 --> 00:35:10.360
So now I can make a goes
to a very small value.
00:35:10.360 --> 00:35:14.120
We fixed the Ts
and the row S. Very
00:35:14.120 --> 00:35:17.790
similar to what we have learned
from the one dimensional case.
00:35:17.790 --> 00:35:20.030
Basically what we
actually found is
00:35:20.030 --> 00:35:21.845
that time in the
one dimensional case
00:35:21.845 --> 00:35:26.890
is that M minus 1 K metrics
become minus T over Rho
00:35:26.890 --> 00:35:29.710
L, partial squared,
partial x squared
00:35:29.710 --> 00:35:33.200
in the one dimensional case.
00:35:33.200 --> 00:35:37.040
And in the two dimensional
case, without working
00:35:37.040 --> 00:35:40.160
through all the detail
of mass, basically
00:35:40.160 --> 00:35:43.340
what we are going to get
is partial square partial T
00:35:43.340 --> 00:35:48.280
square Psi xy--
00:35:48.280 --> 00:35:52.150
It's actually a function
of x and y and the time,
00:35:52.150 --> 00:35:55.730
right, because this is actually
a two dimensional system.
00:35:55.730 --> 00:36:02.660
And this will be equal to
V squared, partial squared,
00:36:02.660 --> 00:36:08.630
partial x squared plus partial
squared, partial y squared,
00:36:08.630 --> 00:36:12.680
Psi xy and T--
00:36:12.680 --> 00:36:14.840
very similar to
what we have done
00:36:14.840 --> 00:36:17.570
for the one dimensional system.
00:36:17.570 --> 00:36:22.130
And of course I can, as you
define this, as del squared.
00:36:22.130 --> 00:36:25.796
And basically what you are
going to get is V squared,
00:36:25.796 --> 00:36:28.730
del squared, Psi x,y,t.
00:36:34.130 --> 00:36:37.430
So basically we again
see this wave equation,
00:36:37.430 --> 00:36:41.990
but this wave equation is now a
two dimensional wave equation.
00:36:41.990 --> 00:36:46.370
And we can also figure out what
will be the V value, right,
00:36:46.370 --> 00:36:48.440
so what will be the velocity?
00:36:48.440 --> 00:36:55.330
The velocity which is going
to be square root of Ts
00:36:55.330 --> 00:37:01.490
over Rho S. This is very
similar what we have done
00:37:01.490 --> 00:37:05.905
for the continuous case, and
in this case, what replaced
00:37:05.905 --> 00:37:10.970
T over Rho L is Ts
over Rho S. Therefore,
00:37:10.970 --> 00:37:15.890
what we actually see that if I
increase the surface tension,
00:37:15.890 --> 00:37:18.380
then the velocity will increase.
00:37:18.380 --> 00:37:24.620
If I decrease the
mass per unit area,
00:37:24.620 --> 00:37:31.070
Rho S, then I will be able to
have a much faster traveling
00:37:31.070 --> 00:37:34.790
wave from this kind of media.
00:37:34.790 --> 00:37:38.130
And what we can actually
immediately also write down
00:37:38.130 --> 00:37:45.876
is that the Psi will be
proportional to A sine Kx
00:37:45.876 --> 00:37:54.530
times x, sine Ky times y,
and Psi omega T plus 5,
00:37:54.530 --> 00:37:59.930
where omega is calculated
from the input Kx and Ky
00:37:59.930 --> 00:38:04.805
for this standing wave solution.
00:38:08.430 --> 00:38:13.080
And very similarly, I
can also argue that--
00:38:13.080 --> 00:38:16.290
in the three dimensional
case I can actually
00:38:16.290 --> 00:38:18.890
follow exactly
the same argument.
00:38:18.890 --> 00:38:20.750
Basically, in the
three dimensional case,
00:38:20.750 --> 00:38:25.510
as well, we already see in
the electromagnetic wave
00:38:25.510 --> 00:38:29.640
discussion, the three
dimensional wave equation can
00:38:29.640 --> 00:38:32.100
be written as partial
squared, partial T squared
00:38:32.100 --> 00:38:37.020
Psi is a function of
x, y, and z and T.
00:38:37.020 --> 00:38:42.700
And this will be equal to
V squared, partial squared,
00:38:42.700 --> 00:38:46.920
partial x squared plus partial
squared, partial y squared
00:38:46.920 --> 00:38:51.300
plus partial squared,
partial V squared--
00:38:51.300 --> 00:39:03.141
Psi is a function of x,y,z,
and T. Any questions so far?
00:39:03.141 --> 00:39:03.640
Nope?
00:39:07.320 --> 00:39:12.640
OK, so everything is
crystal clear, right?
00:39:12.640 --> 00:39:15.570
OK, so this is actually
the animation, which
00:39:15.570 --> 00:39:17.280
I showed you before already.
00:39:17.280 --> 00:39:21.670
So this is actually the
two dimensional vibration
00:39:21.670 --> 00:39:23.070
of membranous.
00:39:23.070 --> 00:39:26.490
So basically the first one
is what I have shown you
00:39:26.490 --> 00:39:31.770
when I choose a very
small K value, which
00:39:31.770 --> 00:39:35.430
only make half of
the sign and which
00:39:35.430 --> 00:39:38.010
match the boundary condition.
00:39:38.010 --> 00:39:41.340
Basically you see that
there are oscillation,
00:39:41.340 --> 00:39:44.460
which you have the
middle part, which
00:39:44.460 --> 00:39:48.480
is either going toward
you or going away from you
00:39:48.480 --> 00:39:49.930
in this continuous system.
00:39:49.930 --> 00:39:55.030
So basically the solution is
actually remarkably the same
00:39:55.030 --> 00:39:57.840
as what we have seen
in the discrete system.
00:39:57.840 --> 00:39:59.810
OK, that's actually
what I wanted to say.
00:39:59.810 --> 00:40:04.590
And also, of course, you
can increase the k value,
00:40:04.590 --> 00:40:10.290
so that you go to the higher
frequency normal mode.
00:40:10.290 --> 00:40:14.240
And you can see that if you
have more and more nodal
00:40:14.240 --> 00:40:21.171
lines, which is actually
the lines describing the--
00:40:21.171 --> 00:40:24.870
the lines which you
actually have no oscillation
00:40:24.870 --> 00:40:26.400
at all on the surface.
00:40:26.400 --> 00:40:29.040
For example, in this
case, the nodal lines,
00:40:29.040 --> 00:40:32.070
as you're passing through
the middle of this figure--
00:40:32.070 --> 00:40:36.570
because all those little mass,
all the other high particles
00:40:36.570 --> 00:40:38.730
are vibrating like crazy.
00:40:38.730 --> 00:40:43.840
But all the particles on
this line, the nodal line,
00:40:43.840 --> 00:40:48.470
they're not at all moving,
because that's actually
00:40:48.470 --> 00:40:49.920
at this position--
00:40:49.920 --> 00:40:53.880
which is having one of the
sine function equal to 0.
00:40:53.880 --> 00:40:57.090
Therefore, no matter what
you do as a function of time,
00:40:57.090 --> 00:41:00.850
how you evolve the system, all
those particle at that line
00:41:00.850 --> 00:41:04.230
will not move at all.
00:41:04.230 --> 00:41:10.470
And this was demonstrated
from this table here.
00:41:10.470 --> 00:41:13.890
It's actually Chladni figures.
00:41:13.890 --> 00:41:17.610
You can see that in a
two dimensional case
00:41:17.610 --> 00:41:21.600
the figures can look
very complicated.
00:41:21.600 --> 00:41:23.670
So basically what
it's showing here
00:41:23.670 --> 00:41:27.150
is that you have a
square plate and it's
00:41:27.150 --> 00:41:34.080
attached to a vibrator,
and basically this vibrator
00:41:34.080 --> 00:41:35.460
can be controlled.
00:41:35.460 --> 00:41:39.390
I can change the frequency
of that vibration.
00:41:39.390 --> 00:41:47.940
When I reach resonance, which
excites one of the normal mode,
00:41:47.940 --> 00:41:52.620
then this plate
will be oscillating
00:41:52.620 --> 00:41:56.440
in a specific pattern.
00:41:56.440 --> 00:41:58.470
And those lines are
actually showing you
00:41:58.470 --> 00:42:02.670
that the plates, which you
have no oscillation at all
00:42:02.670 --> 00:42:04.050
as a function of time.
00:42:04.050 --> 00:42:09.820
Because if I, for example,
turn on this demo again,
00:42:09.820 --> 00:42:12.483
you can see that if
I turn on this demo--
00:42:17.320 --> 00:42:21.430
you can see that all
the sand on the plates
00:42:21.430 --> 00:42:25.430
are vibrating because
now I am oscillating
00:42:25.430 --> 00:42:32.430
this plate by the vibration
generator and the button--
00:42:32.430 --> 00:42:33.720
by the motor and the button.
00:42:33.720 --> 00:42:38.050
And if I change the
oscillation frequency so
00:42:38.050 --> 00:42:43.020
you can see that this frequency
doesn't match with one
00:42:43.020 --> 00:42:46.860
of the normal mode frequency.
00:42:46.860 --> 00:42:50.160
Therefore, they will not
be a lot of activity.
00:42:50.160 --> 00:42:56.590
But if I now change
the frequency,
00:42:56.590 --> 00:43:02.370
so that it matches with one
of the oscillation frequency
00:43:02.370 --> 00:43:05.280
for one of the normal
mode of this system,
00:43:05.280 --> 00:43:11.450
you can see the, oh, some really
cryptic pattern is formed!
00:43:11.450 --> 00:43:16.080
You can see that, oh, it have
a very complicated pattern.
00:43:16.080 --> 00:43:19.810
And if I put my finger
in one of the lines
00:43:19.810 --> 00:43:23.480
here I don't feel the vibration,
but on the other hand,
00:43:23.480 --> 00:43:25.980
if I put my finger
here, I can actually
00:43:25.980 --> 00:43:30.920
feel that there's lot of
vibration at that point.
00:43:30.920 --> 00:43:32.960
I can always change
the frequency
00:43:32.960 --> 00:43:36.000
and see what will happen.
00:43:36.000 --> 00:43:40.110
And then you can see that
now I increase the frequency,
00:43:40.110 --> 00:43:45.509
and now I am actually trying
to excite another mode.
00:43:45.509 --> 00:43:50.440
Now I need some more sand.
00:43:50.440 --> 00:43:55.740
You can see that I randomly
throw sand on this plate,
00:43:55.740 --> 00:43:58.880
and then you can see that
those centered as you bounce it
00:43:58.880 --> 00:44:06.060
around until it sits on
the nodal line, which
00:44:06.060 --> 00:44:09.000
no vibration actually happens.
00:44:09.000 --> 00:44:15.576
OK, so let's go back to one of
the lower frequency mode, which
00:44:15.576 --> 00:44:17.052
we showed you before.
00:44:21.000 --> 00:44:24.720
Now the question is, OK, you
can see this complicated pattern
00:44:24.720 --> 00:44:32.490
almost look ridiculous, can we
actually reproduce this pattern
00:44:32.490 --> 00:44:33.720
by our calculation?
00:44:36.650 --> 00:44:42.740
So we have seen that, OK, I can
conclude that the normal mode
00:44:42.740 --> 00:44:44.090
looks like this, right?
00:44:44.090 --> 00:44:51.009
So therefore, I must
be able to explain
00:44:51.009 --> 00:44:52.550
all those patterns,
which is actually
00:44:52.550 --> 00:44:56.780
shown in this experiment.
00:44:56.780 --> 00:45:00.555
So that's actually what I am
going to do to give you a try.
00:45:04.070 --> 00:45:07.650
So this is a little
demonstration
00:45:07.650 --> 00:45:12.110
which I actually wrote.
00:45:12.110 --> 00:45:17.210
This demonstration actually
has the solution to this two
00:45:17.210 --> 00:45:19.460
dimensional problem.
00:45:19.460 --> 00:45:23.300
And also the boundary
condition is that--
00:45:23.300 --> 00:45:26.600
or say, the condition which,
as you can see on my solution,
00:45:26.600 --> 00:45:32.000
is that I require the center
of the plate to be driven,
00:45:32.000 --> 00:45:35.840
because that's where I
start to vibrate this plate.
00:45:35.840 --> 00:45:39.770
And I drive this plate
up and down to see
00:45:39.770 --> 00:45:42.020
what is going to happen.
00:45:42.020 --> 00:45:44.670
So from this
analytical calculation
00:45:44.670 --> 00:45:49.770
you can see that you expect a
circle in the middle and also
00:45:49.770 --> 00:45:55.170
four lines which actually
cover this circle.
00:45:55.170 --> 00:45:57.560
And also there are
some strange structure
00:45:57.560 --> 00:46:02.360
at the edge of the plate.
00:46:02.360 --> 00:46:05.780
And you can actually
compare this calculation
00:46:05.780 --> 00:46:10.430
to this result. It doesn't
really match perfectly.
00:46:10.430 --> 00:46:12.840
So you can see that there
is some imperfection,
00:46:12.840 --> 00:46:16.310
but you get that
ring in the middle,
00:46:16.310 --> 00:46:20.078
and you do see these
1-2-3-4-5-6-7-8,
00:46:20.078 --> 00:46:24.170
8 lines produced
in this experiment.
00:46:24.170 --> 00:46:28.250
So this experiment is not
perfect because there's a,
00:46:28.250 --> 00:46:30.060
you know, stiffness
of this thing
00:46:30.060 --> 00:46:34.640
and also some energy
dissipation, et cetera.
00:46:34.640 --> 00:46:36.590
But you can see
that, sort of, we
00:46:36.590 --> 00:46:41.030
can actually use our calculation
to explain this pattern!
00:46:41.030 --> 00:46:42.920
That's really cool, right?
00:46:42.920 --> 00:46:49.250
And the advantage is that now I
have this wonderful simulation.
00:46:49.250 --> 00:46:53.000
I can put in all
the crazy numbers,
00:46:53.000 --> 00:46:56.760
and you see that, huh, if
I increase the K value,
00:46:56.760 --> 00:47:02.240
I can really make all kinds
of ridiculous patterns out
00:47:02.240 --> 00:47:03.020
of this.
00:47:03.020 --> 00:47:08.640
And all these things can be kind
of realized by this experiment.
00:47:08.640 --> 00:47:10.050
So you can see
that, for example,
00:47:10.050 --> 00:47:17.960
I can now also turn on
this, and I can actually
00:47:17.960 --> 00:47:25.720
increase the frequency to a very
high frequency, for example.
00:47:25.720 --> 00:47:28.690
Then I can see that,
oh, the pattern really
00:47:28.690 --> 00:47:30.690
becomes much more complicated.
00:47:38.498 --> 00:47:42.330
Now I have a circle and there
are many, many more structures
00:47:42.330 --> 00:47:44.740
which you're seeing in
the surrounding area.
00:47:44.740 --> 00:47:48.380
And of course I can
again increase, increase,
00:47:48.380 --> 00:47:51.180
and see what happens.
00:47:51.180 --> 00:47:54.390
I don't know what is going to
happen because every time I
00:47:54.390 --> 00:47:57.270
do this experiment I
get a different pattern.
00:48:01.450 --> 00:48:07.800
OK, now this seems to be
a very nice frequency.
00:48:07.800 --> 00:48:11.790
It's getting harder and harder.
00:48:11.790 --> 00:48:16.100
You can see that
this is really crazy.
00:48:16.100 --> 00:48:18.510
Holy mackerel, right?
00:48:18.510 --> 00:48:21.010
What the hell is this?
00:48:21.010 --> 00:48:23.880
So you can see that all those
crazy patterns can be created.
00:48:26.440 --> 00:48:28.410
And of course,
during the break, you
00:48:28.410 --> 00:48:33.130
are welcome to come
forward and play with this.
00:48:33.130 --> 00:48:36.180
So you can see that
we can actually
00:48:36.180 --> 00:48:41.710
understand, sort of, the pattern
produced from this experiment.
00:48:41.710 --> 00:48:43.960
That's actually very exciting,
because that's actually
00:48:43.960 --> 00:48:45.290
why we are physicists, right?
00:48:45.290 --> 00:48:48.100
We would like to know why
those patterns are formed,
00:48:48.100 --> 00:48:49.660
and now you know why.
00:48:49.660 --> 00:48:51.670
Those patterns are
formed because there
00:48:51.670 --> 00:48:56.830
are nodal lines in this two
dimensional normal mode modes.
00:48:56.830 --> 00:48:59.440
And the little sands
really love to sit there,
00:48:59.440 --> 00:49:02.050
because you want to sit
in a place which you
00:49:02.050 --> 00:49:04.600
don't have a lot of vibration.
00:49:04.600 --> 00:49:06.420
It's not very
comfortable, right?
00:49:06.420 --> 00:49:10.030
So you sit in the place,
which, hm, vibrates?
00:49:10.030 --> 00:49:10.922
Your problem.
00:49:10.922 --> 00:49:11.880
Vibrate's your problem.
00:49:11.880 --> 00:49:15.740
I sit here where
there is no vibration.
00:49:15.740 --> 00:49:20.260
So that's basically how we
explain these strange figures
00:49:20.260 --> 00:49:21.480
which we can see.
00:49:21.480 --> 00:49:25.270
And just for fun you can
see that I can also generate
00:49:25.270 --> 00:49:27.420
all kinds of craziness.
00:49:27.420 --> 00:49:31.480
You can input all kinds of
different Nx and Ny values,
00:49:31.480 --> 00:49:35.110
and you get all those
wonderful figures for free.
00:49:35.110 --> 00:49:37.600
Maybe we can actually
make some T-shirts
00:49:37.600 --> 00:49:41.260
with all those figures
on the T-shirt, right?
00:49:41.260 --> 00:49:46.790
OK, so we had a lot of fun with
this two dimensional plate.
00:49:46.790 --> 00:49:52.480
How about what will happen
if I have a circular plate?
00:49:52.480 --> 00:49:54.620
What does it do?
00:49:54.620 --> 00:49:58.390
Unfortunately, I would not
be able to solve the two
00:49:58.390 --> 00:50:01.600
dimensional plate
problem in front of you
00:50:01.600 --> 00:50:05.920
because that will give you a
Bessel function, which is not
00:50:05.920 --> 00:50:09.410
the end of the world, but that's
actually kind of complicated.
00:50:09.410 --> 00:50:12.790
If I put it in mid-term
exam, that's actually not
00:50:12.790 --> 00:50:15.520
very encouraging, right?
00:50:15.520 --> 00:50:17.980
But I can actually tell you
what will be the solution.
00:50:17.980 --> 00:50:19.780
The solution will be
a Bessel function.
00:50:19.780 --> 00:50:24.400
Basically you will have a
lot of ring-like structures
00:50:24.400 --> 00:50:27.190
if I have a circular plate.
00:50:27.190 --> 00:50:30.460
And I can actually do an
experiment which actually shows
00:50:30.460 --> 00:50:35.740
you the behavior of the
circular boundary condition
00:50:35.740 --> 00:50:39.370
and see what kind of
pattern can we see.
00:50:39.370 --> 00:50:45.790
So here I have a kind of
complicated experiment.
00:50:45.790 --> 00:50:48.760
So here I have
this ring, which I
00:50:48.760 --> 00:50:56.580
would like to produce
some film on this ring.
00:50:56.580 --> 00:50:59.140
So see if I am successful.
00:50:59.140 --> 00:51:00.770
Kind of.
00:51:00.770 --> 00:51:01.620
OK.
00:51:01.620 --> 00:51:12.100
Now I can put this a soft
film in front of the speaker.
00:51:12.100 --> 00:51:16.460
I can actually
oscillate this thing--
00:51:16.460 --> 00:51:18.527
membranes by the speaker.
00:51:18.527 --> 00:51:20.360
Oops, don't want to
destroy everything here.
00:51:24.510 --> 00:51:28.910
All right, so now I can turn
on this, so that we have light.
00:51:28.910 --> 00:51:32.990
And of course I will turn
on the signal generator,
00:51:32.990 --> 00:51:34.580
so that I can hear--
00:51:34.580 --> 00:51:39.020
I can actually start to
vibrate the membranes.
00:51:39.020 --> 00:51:47.185
Before I do that, I have to turn
everything off, hide images.
00:51:47.185 --> 00:51:49.870
All right, I hope you
can see something.
00:51:49.870 --> 00:51:50.370
Can you?
00:51:53.690 --> 00:51:55.420
Can you see something on the--
00:51:58.210 --> 00:52:05.030
it's kind of difficult to see
it, but that should be there.
00:52:05.030 --> 00:52:10.140
OK, now I can turn
on this speaker,
00:52:10.140 --> 00:52:13.040
and you can see that there
are some patterns which
00:52:13.040 --> 00:52:16.160
it's probably difficult to see.
00:52:16.160 --> 00:52:18.360
Kind of see, right?
00:52:18.360 --> 00:52:20.560
There are rings.
00:52:20.560 --> 00:52:23.336
You can see it on the speaker.
00:52:23.336 --> 00:52:27.710
So you can see that now I have
one, two, three-- three rings,
00:52:27.710 --> 00:52:28.850
right?
00:52:28.850 --> 00:52:33.950
Because I couldn't turn off
the light, which is actually
00:52:33.950 --> 00:52:37.220
emitted from the sun, right?
00:52:37.220 --> 00:52:39.080
So I cannot turn off sun.
00:52:39.080 --> 00:52:42.130
Therefore, you can
barely see this figure.
00:52:42.130 --> 00:52:45.470
So we shall explain the
result of this experiment.
00:52:45.470 --> 00:52:51.540
And you can see that if
I increase the frequency,
00:52:51.540 --> 00:52:54.270
according to the solution
from Bessel function,
00:52:54.270 --> 00:52:58.312
you will see more
rings got excited.
00:52:58.312 --> 00:53:00.270
So you can see that now
I have one, two, three,
00:53:00.270 --> 00:53:01.740
four-- four rings.
00:53:01.740 --> 00:53:04.542
And of course I can continue
to increase, increase,
00:53:04.542 --> 00:53:06.102
and increase.
00:53:06.102 --> 00:53:10.470
And you will see that there
are even more rings produced.
00:53:10.470 --> 00:53:13.280
Essentially what I'm
doing is actually really
00:53:13.280 --> 00:53:18.500
trying to vibrate and excite
one of the normal mode
00:53:18.500 --> 00:53:21.530
by this loud speaker.
00:53:21.530 --> 00:53:23.420
And you can actually
kind of see--
00:53:23.420 --> 00:53:26.270
I hope you can kind of see it.
00:53:26.270 --> 00:53:28.950
If you can still
see it, that means
00:53:28.950 --> 00:53:32.145
you need to check your eye
because the membranes is
00:53:32.145 --> 00:53:32.645
broken.
00:53:36.560 --> 00:53:39.800
OK, so I think you
sort of get this idea,
00:53:39.800 --> 00:53:45.470
and I'm going to turn off
this wonderful machine
00:53:45.470 --> 00:53:50.250
and go back to the lecture.
00:53:50.250 --> 00:53:56.570
So this experiment is kind of
hard to reproduce in your study
00:53:56.570 --> 00:53:58.010
room, right?
00:53:58.010 --> 00:54:00.200
I think everybody will agree.
00:54:00.200 --> 00:54:03.170
And there's another
one which is actually
00:54:03.170 --> 00:54:05.300
kind of easy to
reproduce, which I
00:54:05.300 --> 00:54:07.730
will encourage you to try it--
00:54:07.730 --> 00:54:08.990
so if you have time.
00:54:08.990 --> 00:54:11.500
So this is from Jake.
00:54:11.500 --> 00:54:14.510
He sent me this
wonderful video when
00:54:14.510 --> 00:54:16.700
I was teaching the 8.03 class.
00:54:16.700 --> 00:54:20.780
They found that they could
excite two dimensional waves
00:54:20.780 --> 00:54:23.470
in this way.
00:54:23.470 --> 00:54:25.400
Can you see it?
00:54:25.400 --> 00:54:27.410
It's wonderful.
00:54:27.410 --> 00:54:33.620
You can see there are very high
frequency oscillation, which
00:54:33.620 --> 00:54:37.640
actually excite these
two dimensional wave.
00:54:37.640 --> 00:54:41.210
And you can see that lots,
and lots, and lots of rings
00:54:41.210 --> 00:54:43.280
are excited.
00:54:43.280 --> 00:54:44.990
And then you can
see very clearly
00:54:44.990 --> 00:54:50.240
from this simple
experiment, what you really
00:54:50.240 --> 00:54:53.420
need is a cup of water.
00:54:53.420 --> 00:54:57.740
And you rub it against the
surface of a table, then
00:54:57.740 --> 00:55:01.610
you'll be able to excite all
the crazy patterns, which
00:55:01.610 --> 00:55:07.510
you can actually see from
this two dimensional system
00:55:07.510 --> 00:55:11.180
and with two dimensional
boundary conditions.
00:55:11.180 --> 00:55:13.580
OK, so we will
take a five minute
00:55:13.580 --> 00:55:18.560
break before we enter the
next part of the discussion.
00:55:18.560 --> 00:55:20.280
And we come back at 35.
00:55:26.940 --> 00:55:29.250
OK, welcome back, everybody.
00:55:29.250 --> 00:55:33.180
So what I'm going to do now
is to continue the discussion,
00:55:33.180 --> 00:55:37.550
the one we actually got
started, of the two dimensional
00:55:37.550 --> 00:55:39.600
and three dimensional system.
00:55:39.600 --> 00:55:45.030
And we have actually
studied the behavior
00:55:45.030 --> 00:55:48.520
of standing wave,
or normal mode,
00:55:48.520 --> 00:55:51.220
for this two dimensional system.
00:55:51.220 --> 00:55:55.080
And what I am going to do
is discuss with you, a two
00:55:55.080 --> 00:55:56.900
dimensional progressive wave.
00:56:06.470 --> 00:56:11.120
So I will stick to a
really simple example,
00:56:11.120 --> 00:56:12.480
which are plane waves.
00:56:18.720 --> 00:56:24.010
OK, so in the case of plane
waves, which we discussed when
00:56:24.010 --> 00:56:27.520
we actually discussed
the EM waves,
00:56:27.520 --> 00:56:31.100
you have the following
functional form.
00:56:31.100 --> 00:56:34.600
Psi is a function
of r and the t.
00:56:34.600 --> 00:56:41.160
And this will be equal
to A exponential i.
00:56:41.160 --> 00:56:44.770
The k is a vector
now, and it's pointing
00:56:44.770 --> 00:56:50.570
to the direction of the
propagation of this plan wave.
00:56:50.570 --> 00:56:57.080
And this k is dot with r
vector minus omega T, which
00:56:57.080 --> 00:57:01.310
is the oscillation
frequency-- angular frequency.
00:57:01.310 --> 00:57:04.670
And evaluated at
a specific time.
00:57:04.670 --> 00:57:09.590
And this is expression
actually describes a plane wave
00:57:09.590 --> 00:57:14.540
where the direction
of propagation
00:57:14.540 --> 00:57:18.110
is described by this k vector.
00:57:18.110 --> 00:57:22.820
And of course you can
actually have the wave front,
00:57:22.820 --> 00:57:27.320
which is actually the peak
position of this plain wave.
00:57:27.320 --> 00:57:31.220
And the distance between
the peak position--
00:57:31.220 --> 00:57:33.710
so if you can imagine
that this is like this.
00:57:44.030 --> 00:57:48.530
So if you look at the
distance between peak position
00:57:48.530 --> 00:57:51.920
that will give you the
wavelengths, right?
00:57:51.920 --> 00:57:54.530
The wavelengths,
now that will be
00:57:54.530 --> 00:57:58.310
equal to 2 pi
divided by k, right?
00:57:58.310 --> 00:58:03.330
In this case, it's the
length of this k vector.
00:58:03.330 --> 00:58:06.560
Just a reminder about
what we introduced
00:58:06.560 --> 00:58:09.170
in the previous lecture.
00:58:09.170 --> 00:58:13.520
And we were using this to
describe electromagnetic wave
00:58:13.520 --> 00:58:17.570
and such a kind of
expression can be also
00:58:17.570 --> 00:58:22.880
be used to describe
sound waves and also
00:58:22.880 --> 00:58:28.010
vibration on the membranes,
et cetera, progressive waves.
00:58:28.010 --> 00:58:33.230
So if there are no other
medium like what we actually
00:58:33.230 --> 00:58:35.450
have in this slide--
00:58:35.450 --> 00:58:37.460
so we have nothing else.
00:58:37.460 --> 00:58:42.410
I have a membrane with
a surface tension Ts,
00:58:42.410 --> 00:58:47.360
and Rho S is the
mass per unit area.
00:58:47.360 --> 00:58:50.080
Then basically, this
progressing wave
00:58:50.080 --> 00:58:53.540
is going to be traveling
at the speed of v,
00:58:53.540 --> 00:58:57.830
which is equal to square
root of Ts over Rho S,
00:58:57.830 --> 00:59:02.600
and I can actually define that
to be some constant c divided
00:59:02.600 --> 00:59:03.860
by n.
00:59:03.860 --> 00:59:13.100
So c is some constant,
and m is another constant
00:59:13.100 --> 00:59:16.610
which actually, the ratio
c and n is equal to v.
00:59:16.610 --> 00:59:20.800
And I will need that expression
later, only later, not now.
00:59:20.800 --> 00:59:26.030
If I have nothing else and
that this system actually
00:59:26.030 --> 00:59:28.850
filled the whole universe,
then what is going to happen
00:59:28.850 --> 00:59:31.160
is that this progressing
wave is going
00:59:31.160 --> 00:59:34.600
to be propagating, propagating,
propagating, propagating.
00:59:34.600 --> 00:59:39.680
Nothing will change until
the edge of the universe.
00:59:39.680 --> 00:59:43.700
It doesn't actually
introduce any excitement.
00:59:43.700 --> 00:59:45.890
So that's what we
have already learned
00:59:45.890 --> 00:59:52.010
from when we have discussed
electromagnetic interaction,
00:59:52.010 --> 00:59:54.290
and now the same
expression can also
00:59:54.290 --> 01:00:00.080
be used for the description
of the membranes.
01:00:00.080 --> 01:00:04.850
And then now to make this
problem more exciting,
01:00:04.850 --> 01:00:08.410
what I'm going to do is
to introduce a boundary.
01:00:08.410 --> 01:00:12.340
So the boundary is in
the middle of this slide.
01:00:12.340 --> 01:00:17.690
And I will assume that
the horizontal direction
01:00:17.690 --> 01:00:22.070
to be x equal to 0--
01:00:22.070 --> 01:00:27.350
the horizontal direction to be
in x direction and the boundary
01:00:27.350 --> 01:00:29.210
is at x equal to 0.
01:00:32.330 --> 01:00:39.130
And when you pass
this boundary, there's
01:00:39.130 --> 01:00:44.500
another kind of material
with surface tension Ts prime
01:00:44.500 --> 01:00:50.920
and slightly different mass
per unit area, your s prime.
01:00:50.920 --> 01:00:54.360
Based on the expression
we got for the velocity
01:00:54.360 --> 01:00:56.780
we will be able to
conclude that v prime will
01:00:56.780 --> 01:01:02.650
be equal to square root of T
prime S divided by Rho S prime.
01:01:02.650 --> 01:01:06.580
And that will be equal
to c over n prime.
01:01:06.580 --> 01:01:10.630
And c is the same constant which
I used for the left hand side
01:01:10.630 --> 01:01:12.260
system.
01:01:12.260 --> 01:01:16.900
And n, later, you will realize
that that's a refraction
01:01:16.900 --> 01:01:20.710
index in a discussion.
01:01:20.710 --> 01:01:24.580
So the question which
I would like to ask
01:01:24.580 --> 01:01:29.820
is, OK, now I have a
prime wave propagating
01:01:29.820 --> 01:01:31.200
in the first system.
01:01:31.200 --> 01:01:35.020
And it met a boundary,
and the question
01:01:35.020 --> 01:01:41.020
is what will happen when I
have the incident wave coming
01:01:41.020 --> 01:01:42.160
into the system?
01:01:44.830 --> 01:01:50.920
So before that, I also need
to write down the dispersion
01:01:50.920 --> 01:01:51.940
relation, right.
01:01:51.940 --> 01:01:55.090
So dispersion relation
can be attempted
01:01:55.090 --> 01:02:01.660
by plugging in a normal
mode Psi function
01:02:01.660 --> 01:02:03.730
into the wave equation.
01:02:03.730 --> 01:02:06.700
So what I can
immediately obtain is
01:02:06.700 --> 01:02:10.020
that the dispersion
relation, omega squared
01:02:10.020 --> 01:02:16.990
is equal to V squared, Kx
squared, plus Ky squared.
01:02:16.990 --> 01:02:19.390
You can actually
check this expression
01:02:19.390 --> 01:02:24.430
by plugging in this function
into the two dimensional wave
01:02:24.430 --> 01:02:29.170
equation, and you will
get that expression, OK?
01:02:29.170 --> 01:02:33.160
And that means omega
cannot be arbitrary number.
01:02:33.160 --> 01:02:36.640
It's as you decided
by Kx and the Ky.
01:02:36.640 --> 01:02:41.410
Or say, if omega is the side
and one of the k is the side,
01:02:41.410 --> 01:02:45.490
then the third number, for
example in this case, Kx,
01:02:45.490 --> 01:02:48.940
is as you decided by
the dispersion relation
01:02:48.940 --> 01:02:51.790
which we have here.
01:02:51.790 --> 01:02:56.160
So, coming back to the original
problem we are posting,
01:02:56.160 --> 01:03:01.860
I have, now, the incident
wave coming into this system.
01:03:01.860 --> 01:03:06.390
I would like to know what
will happen at the boundary
01:03:06.390 --> 01:03:12.070
when I have two systems with a
left hand side propagating at--
01:03:12.070 --> 01:03:14.280
the speed of the
propagation is v,
01:03:14.280 --> 01:03:17.400
and right hand side's speed
of propagation is v prime.
01:03:17.400 --> 01:03:18.450
What is going to happen?
01:03:21.990 --> 01:03:28.810
Assume my guess that I am
going to get a refractive wave
01:03:28.810 --> 01:03:31.210
and a transmitted wave.
01:03:31.210 --> 01:03:34.330
So that's based on what we
have learned from the one
01:03:34.330 --> 01:03:36.370
dimensional system.
01:03:36.370 --> 01:03:40.580
If I call this the left hand
side, and call the right hand
01:03:40.580 --> 01:03:43.710
side system right hand--
01:03:43.710 --> 01:03:44.992
the right hand system, r.
01:03:47.860 --> 01:03:56.150
So I can write down the wave
function Psi L describing
01:03:56.150 --> 01:03:57.680
the left hand side.
01:03:57.680 --> 01:04:05.140
This will be equal to A
exponential of ik dot r
01:04:05.140 --> 01:04:10.070
minus omega T. This is
actually the incident wave--
01:04:19.750 --> 01:04:22.500
describing this incident wave.
01:04:22.500 --> 01:04:25.380
And as you might
guess, there should be
01:04:25.380 --> 01:04:27.660
some kind of refraction, right?
01:04:27.660 --> 01:04:31.710
So once this wave
actually passed
01:04:31.710 --> 01:04:35.070
through the boundary,
or touch the boundary,
01:04:35.070 --> 01:04:37.470
there should be some kind
of refraction, right?
01:04:37.470 --> 01:04:39.970
So the refraction, I can
actually write it down
01:04:39.970 --> 01:04:44.490
in this form as sum
over alpha, r alpha
01:04:44.490 --> 01:04:48.660
is actually the coefficient
over amplitude as function
01:04:48.660 --> 01:04:51.440
of the normal modes--
01:04:51.440 --> 01:04:58.150
as a function of the
progressing wave number,
01:04:58.150 --> 01:04:59.590
which I have shown.
01:04:59.590 --> 01:05:01.650
And I can actually
sum over all kinds
01:05:01.650 --> 01:05:05.110
of progressing wave numbers.
01:05:05.110 --> 01:05:12.990
Exponential ik alpha
times r minus omega T.
01:05:12.990 --> 01:05:18.805
So this is a general
form of refracting wave.
01:05:18.805 --> 01:05:23.820
k alpha is describing
the direction
01:05:23.820 --> 01:05:25.950
of the individual
refractive wave,
01:05:25.950 --> 01:05:30.899
and alpha is labeling the
individual refractive wave.
01:05:30.899 --> 01:05:32.940
But I don't know what will
be the functional form
01:05:32.940 --> 01:05:35.230
for the k alpha for the moment.
01:05:35.230 --> 01:05:40.380
So therefore, I try to sum
over all the possible alpha.
01:05:40.380 --> 01:05:43.590
And I would like
to figure out what
01:05:43.590 --> 01:05:47.700
will be the allowed alpha
by matching the boundary
01:05:47.700 --> 01:05:48.850
condition.
01:05:48.850 --> 01:05:52.460
So in short, the
right hand side turn
01:05:52.460 --> 01:05:54.972
essentially is actually
describing the refractive wave.
01:06:00.540 --> 01:06:04.710
And finally, passing through
this boundary condition,
01:06:04.710 --> 01:06:07.500
let's look at the
right hand side.
01:06:07.500 --> 01:06:12.990
Right hand side, Psi
r, is going to be
01:06:12.990 --> 01:06:20.880
sum over beta on the
transmission coefficients tau
01:06:20.880 --> 01:06:28.570
beta, which is the original
amplitude, exponential of i,
01:06:28.570 --> 01:06:36.790
k beta times r minus
omega T. So again I
01:06:36.790 --> 01:06:39.880
don't know what
will be the behavior
01:06:39.880 --> 01:06:41.080
of the transmitted wave.
01:06:41.080 --> 01:06:44.380
Therefore, I have summed
over all the possible values.
01:06:44.380 --> 01:06:46.547
And this is actually
the functional form
01:06:46.547 --> 01:06:47.588
for the transmitted wave.
01:06:56.750 --> 01:07:04.190
I also know that k alpha
vector squared will
01:07:04.190 --> 01:07:10.550
be equal to omega
squared Rho s over Ts,
01:07:10.550 --> 01:07:15.890
and this will be equal to
omega squared v squared,
01:07:15.890 --> 01:07:19.760
because of the dispersion
relation in the left hand side.
01:07:19.760 --> 01:07:24.110
So basically, if you look at
the left hand side dispersion
01:07:24.110 --> 01:07:31.680
relation, the length
squared of this k vector
01:07:31.680 --> 01:07:34.850
will be equal to omega squared
times v squared, right?
01:07:34.850 --> 01:07:37.660
This is just a
dispersion relation
01:07:37.660 --> 01:07:41.070
of a non-dispersive medium.
01:07:41.070 --> 01:07:45.720
And also, I can actually
figure out what will be the--
01:07:45.720 --> 01:07:49.200
allowed length for the k theta.
01:07:49.200 --> 01:07:54.810
So the k theta squared will
be equal to omega squared
01:07:54.810 --> 01:08:00.110
v prime square, because this
progressing wave is actually
01:08:00.110 --> 01:08:02.580
the transmitted
wave, is actually
01:08:02.580 --> 01:08:06.680
traveling in a second medium.
01:08:10.620 --> 01:08:13.180
So look at what
we have done here.
01:08:13.180 --> 01:08:16.270
So we have an incident wave.
01:08:16.270 --> 01:08:18.729
We will wonder, then,
what is going to happen.
01:08:18.729 --> 01:08:21.569
Our physics intuition
tells me that, you
01:08:21.569 --> 01:08:25.000
must get a refracting
wave, oscillation frequency
01:08:25.000 --> 01:08:25.979
should be the same.
01:08:25.979 --> 01:08:27.600
Otherwise, as a
function over time,
01:08:27.600 --> 01:08:30.930
you cannot match the left
hand and right hand side.
01:08:30.930 --> 01:08:35.170
And you also get a
transmitted wave.
01:08:35.170 --> 01:08:39.180
But I'm now in trouble
because I have so many turns.
01:08:39.180 --> 01:08:42.090
I'm summing over alpha
infinite number of turns,
01:08:42.090 --> 01:08:46.220
and I don't know what will be
the coefficient for the r alpha
01:08:46.220 --> 01:08:48.770
and the tau beta, which
are the transmission
01:08:48.770 --> 01:08:52.380
coefficient and then
refraction coefficients.
01:08:52.380 --> 01:08:56.550
So what I need to do,
as you might guess,
01:08:56.550 --> 01:09:00.380
is to use the
boundary condition.
01:09:00.380 --> 01:09:02.770
So now I am writing
down, already,
01:09:02.770 --> 01:09:04.170
the general expression.
01:09:04.170 --> 01:09:06.600
Now I'm going to use
the boundary condition
01:09:06.600 --> 01:09:11.910
to actually limit the choice of
the possible k alpha and the k
01:09:11.910 --> 01:09:12.810
beta.
01:09:12.810 --> 01:09:14.710
What is actually the
boundary condition?
01:09:21.580 --> 01:09:28.729
The boundary conditions
are that at x equal to 0--
01:09:28.729 --> 01:09:32.180
that's actually at the
position of this line--
01:09:32.180 --> 01:09:34.335
the membranes doesn't break.
01:09:37.750 --> 01:09:42.850
Otherwise, suddenly
the membranes break,
01:09:42.850 --> 01:09:46.485
and this is the end of
the discussion, right?
01:09:46.485 --> 01:09:48.430
Like, what we have
done before, right?
01:09:48.430 --> 01:09:50.229
So the membranes
doesn't break, so
01:09:50.229 --> 01:09:53.620
that the propagation
can continue.
01:09:53.620 --> 01:09:56.420
So what does that mean?
01:09:56.420 --> 01:10:02.620
This means that if I
evaluate Psi L and Psi
01:10:02.620 --> 01:10:08.650
r at x equal to 0, Psi 0, y, t.
01:10:11.500 --> 01:10:17.890
The left hand side will be equal
to A exponential i, Ky times y
01:10:17.890 --> 01:10:26.360
minus omega T, plus summing over
all possible alpha, r alpha,
01:10:26.360 --> 01:10:37.780
A exponential i, K alpha y
times y, minus omega T. This
01:10:37.780 --> 01:10:42.400
is the incident wave transmitted
wave evaluated at the left hand
01:10:42.400 --> 01:10:46.230
side, which is
the upper formula.
01:10:46.230 --> 01:10:48.990
And that will be equal
to the right hand
01:10:48.990 --> 01:10:52.320
side, which is containing
only the transmitted wave.
01:10:52.320 --> 01:10:55.150
So basically you
have summing over
01:10:55.150 --> 01:11:02.790
beta, tau beta, A
exponential i, k beta,
01:11:02.790 --> 01:11:08.440
y times r minus omega T.
01:11:08.440 --> 01:11:13.650
And this expression,
this boundary condition,
01:11:13.650 --> 01:11:17.600
should hold true for
all the possible y,
01:11:17.600 --> 01:11:19.920
right, because the
boundary condition
01:11:19.920 --> 01:11:22.530
is valid at x equal to 0.
01:11:22.530 --> 01:11:26.670
I didn't specify the value of y.
01:11:26.670 --> 01:11:30.090
So therefore I can actually
put in all the possible-- oh,
01:11:30.090 --> 01:11:31.830
this should be y.
01:11:31.830 --> 01:11:32.700
Sorry for that.
01:11:32.700 --> 01:11:37.590
I can actually vary the y, and
I will figure out that, ah,
01:11:37.590 --> 01:11:42.680
if I have Ky not equal
to k alpha y, that
01:11:42.680 --> 01:11:46.920
means the wavelengths
of the refractive wave
01:11:46.920 --> 01:11:49.590
and the incident wave
will be different.
01:11:49.590 --> 01:11:54.750
If I have Ky not equal
to beta y that means
01:11:54.750 --> 01:11:57.150
the transmitted
wavelengths is going
01:11:57.150 --> 01:12:00.330
to be different from
the incident wave.
01:12:00.330 --> 01:12:04.110
That means, no matter what
I do as a boundary of y,
01:12:04.110 --> 01:12:05.455
the membranes will break.
01:12:08.290 --> 01:12:13.840
Therefore, in order to
make this equation valid,
01:12:13.840 --> 01:12:18.810
the only choice is
that when k alpha y
01:12:18.810 --> 01:12:23.840
will be equal to k
beta y and equal to Ky.
01:12:23.840 --> 01:12:28.000
So that means the wavelengths
projected in the y direction
01:12:28.000 --> 01:12:33.550
should be equal for the
incident wave, transmitted wave,
01:12:33.550 --> 01:12:34.870
and the refractive wave.
01:12:34.870 --> 01:12:39.390
Otherwise, as you always move a
little bit in the y direction,
01:12:39.390 --> 01:12:42.250
the membranes will break.
01:12:42.250 --> 01:12:47.770
So that's actually the condition
which you can actually get.
01:12:47.770 --> 01:12:53.810
And the interesting thing is
that, based on this expression,
01:12:53.810 --> 01:12:57.130
k alpha, the length
of the k alpha,
01:12:57.130 --> 01:13:01.960
and the length of
the k beta is fixed.
01:13:01.960 --> 01:13:07.850
And I also know what will be the
component for the y direction.
01:13:07.850 --> 01:13:15.820
Therefore, that means the x
direction Psi's for that k
01:13:15.820 --> 01:13:21.310
alpha x and the k beta
x are also fixed because
01:13:21.310 --> 01:13:24.750
of the dispersion relation.
01:13:24.750 --> 01:13:29.680
So that immediately brings
me to this conclusion
01:13:29.680 --> 01:13:37.660
that basically k alpha x will
be equal to minus omega squared
01:13:37.660 --> 01:13:41.230
over v squared minus
Ky squared, and that
01:13:41.230 --> 01:13:45.280
will be equal to minus Kx.
01:13:45.280 --> 01:13:48.730
So this is the x component
of the refractive wave.
01:13:48.730 --> 01:13:51.520
And the transmitting
wave, k beta x,
01:13:51.520 --> 01:13:56.290
will be equal to square root
of omega squared over v squared
01:13:56.290 --> 01:13:59.470
minus Ky squared.
01:13:59.470 --> 01:14:04.450
If I draw, visualize
the relative direction
01:14:04.450 --> 01:14:07.875
of all the three
components, basically, this
01:14:07.875 --> 01:14:13.840
is essentially the direction
of the incident wave, k,
01:14:13.840 --> 01:14:17.370
and the incident angle is theta.
01:14:17.370 --> 01:14:20.410
And from this expression,
you see that the Ky
01:14:20.410 --> 01:14:23.380
is equal to k alpha y.
01:14:23.380 --> 01:14:26.180
Therefore, you have
a refractive wave.
01:14:26.180 --> 01:14:28.910
But actually only the x
direction has changed sides.
01:14:28.910 --> 01:14:31.430
Therefore, you have
a refractive wave
01:14:31.430 --> 01:14:37.060
with exactly the same angle
as the incident angle theta.
01:14:37.060 --> 01:14:41.420
The refraction angle
will be zeta as well.
01:14:41.420 --> 01:14:44.380
And that there will
be a transmitted wave
01:14:44.380 --> 01:14:46.330
with theta prime.
01:14:46.330 --> 01:14:51.430
And this is essentially the
direction of the k prime.
01:14:51.430 --> 01:14:57.580
And the interesting thing
is that the projection
01:14:57.580 --> 01:15:01.210
toward the y direction,
that k prime y,
01:15:01.210 --> 01:15:05.200
has to be equal
to the progression
01:15:05.200 --> 01:15:12.460
of the original incident
wave in the y direction.
01:15:12.460 --> 01:15:17.710
So that means I will be
able to conclude that--
01:15:17.710 --> 01:15:19.400
the y components are the same.
01:15:19.400 --> 01:15:23.350
Therefore, I can
conclude that k sine
01:15:23.350 --> 01:15:28.860
theta will be equal to k
prime sine theta prime.
01:15:31.610 --> 01:15:34.000
I'm kind of running out of time.
01:15:34.000 --> 01:15:38.980
And if I define, already
as I defined here,
01:15:38.980 --> 01:15:41.980
velocity is equal to
c over n, and the v
01:15:41.980 --> 01:15:46.840
prime is equal to c over n
prime, what I can immediately
01:15:46.840 --> 01:15:49.870
conclude is that--
01:15:49.870 --> 01:15:51.910
give me one more minute--
01:15:51.910 --> 01:15:57.760
is that if I have n equal
to c over v and the n
01:15:57.760 --> 01:16:00.370
prime is equal to
c over v prime,
01:16:00.370 --> 01:16:03.940
I can conclude that
sine theta will
01:16:03.940 --> 01:16:08.280
be equal to n prime
sine theta prime.
01:16:08.280 --> 01:16:12.520
Does this look familiar to you?
01:16:12.520 --> 01:16:15.610
This is essentially Snell's law.
01:16:15.610 --> 01:16:19.810
How many of you haven't
heard about Snell's law.
01:16:19.810 --> 01:16:21.250
There were a few before.
01:16:21.250 --> 01:16:22.590
Yeah, OK.
01:16:22.590 --> 01:16:23.900
No problem at all.
01:16:23.900 --> 01:16:25.370
Then you learned it.
01:16:25.370 --> 01:16:31.310
So that means if I have two
kinds of systems in my hand,
01:16:31.310 --> 01:16:38.000
and I will be able to relate
the transmitted wave according
01:16:38.000 --> 01:16:40.370
to what I have in
the incident wave.
01:16:40.370 --> 01:16:43.430
And you can see
that Snell's law--
01:16:43.430 --> 01:16:49.100
which were famous for the
discussion of optics--
01:16:49.100 --> 01:16:53.270
and here, I have no
knowledge about optics
01:16:53.270 --> 01:16:55.520
or electromagnetic waves.
01:16:55.520 --> 01:16:59.810
So in short, what I want to
tell you is that, we have just
01:16:59.810 --> 01:17:07.730
proved two of the most important
laws of the geometrical optics,
01:17:07.730 --> 01:17:12.680
the refraction angle is equal to
incident angle and the Snell's
01:17:12.680 --> 01:17:16.520
law without using any
information about the dynamics.
01:17:16.520 --> 01:17:21.350
That means all those laws are
coming from purely boundary
01:17:21.350 --> 01:17:23.850
condition and the waves.
01:17:23.850 --> 01:17:26.700
Therefore, you will
expect that this
01:17:26.700 --> 01:17:31.370
will work for water wave, sound
wave, electromagnetic wave, et
01:17:31.370 --> 01:17:31.870
cetera.
01:17:31.870 --> 01:17:34.900
O So we will continue
the discussion next time.
01:17:34.900 --> 01:17:37.860
Thanks for the attention.
01:17:37.860 --> 01:17:40.496
And if you have any
questions, let me know.
01:17:40.496 --> 01:17:41.120
I will be here.
01:17:46.940 --> 01:17:48.020
Hello, everybody.
01:17:48.020 --> 01:17:50.570
We are going to show you
a demonstration, a really
01:17:50.570 --> 01:17:52.790
nice one.
01:17:52.790 --> 01:17:56.040
It consists of the
following setup.
01:17:56.040 --> 01:17:59.920
So basically I'm going
to place some film here.
01:17:59.920 --> 01:18:04.190
And then behind that
there's a loud speaker,
01:18:04.190 --> 01:18:08.540
which I use as a signal
generator and to actually
01:18:08.540 --> 01:18:09.770
produce sound wave.
01:18:09.770 --> 01:18:13.850
And this sound wave is going
to oscillate the soft film,
01:18:13.850 --> 01:18:17.490
and then you are going
to see the oscillation,
01:18:17.490 --> 01:18:21.470
or the normal mode's
pattern, on the screen.
01:18:21.470 --> 01:18:25.130
OK, so that is
actually the setup
01:18:25.130 --> 01:18:27.650
which we can actually
demonstrate to you two
01:18:27.650 --> 01:18:31.020
dimensional normal modes.
01:18:31.020 --> 01:18:33.170
So the first thing
which I am going to do
01:18:33.170 --> 01:18:35.676
is to produce a soft film.
01:18:38.430 --> 01:18:43.560
Now I am going to put it
back into this setup here.
01:18:43.560 --> 01:18:46.218
You should be able to see
the pattern on the screen.
01:18:50.800 --> 01:18:55.470
Then I am going to turn on
the sound wave generator.
01:18:55.470 --> 01:19:01.110
You can see, immediately,
that the pattern on the screen
01:19:01.110 --> 01:19:03.960
changed because of
the sound wave trying
01:19:03.960 --> 01:19:07.080
to oscillate the soft film.
01:19:07.080 --> 01:19:09.660
You can see it
directly from here,
01:19:09.660 --> 01:19:16.230
but it actually looks much
more prominent on the screen.
01:19:16.230 --> 01:19:22.080
And now what am I going to
do is change the frequency
01:19:22.080 --> 01:19:23.400
of the sound wave.
01:19:23.400 --> 01:19:25.350
And you can see
that I'm changing it
01:19:25.350 --> 01:19:26.730
to a higher frequency.
01:19:29.460 --> 01:19:32.940
And you can see that there is
a more and more complicated
01:19:32.940 --> 01:19:36.080
pattern formed on the screen.
01:19:36.080 --> 01:19:42.560
That is because I'm now exciting
higher and higher frequency
01:19:42.560 --> 01:19:43.480
normal modes.
01:19:46.930 --> 01:19:51.160
And you can see that now I can
actually continue and increase
01:19:51.160 --> 01:19:56.424
the frequency.
01:19:56.424 --> 01:19:57.340
And you can see that--
01:20:01.480 --> 01:20:04.620
now we can see that the
pattern becomes really,
01:20:04.620 --> 01:20:07.680
really infinitely complicated.
01:20:07.680 --> 01:20:10.620
You can see this
grid developing.
01:20:10.620 --> 01:20:14.460
And then you can see
that eventually that's
01:20:14.460 --> 01:20:18.330
basically two sine functions
multiply each other.
01:20:18.330 --> 01:20:20.405
One sine function is
in the x direction.
01:20:20.405 --> 01:20:23.580
The other one is in the
y direction-- horizontal
01:20:23.580 --> 01:20:25.300
and the vertical direction.
01:20:25.300 --> 01:20:28.230
And you can see a
really beautiful pattern
01:20:28.230 --> 01:20:34.320
forming due to the solution
we derived during the lecture.
01:20:34.320 --> 01:20:40.580
And the higher frequency I
go, I can see more and more
01:20:40.580 --> 01:20:46.800
complicating patterns,
many more lines
01:20:46.800 --> 01:20:53.690
developing on the screen due to
oscillation of the soft film.
01:20:53.690 --> 01:20:55.810
You can see now we
have even more lines.
01:20:59.136 --> 01:21:00.760
And it's actually
getting more and more
01:21:00.760 --> 01:21:04.075
difficult to see the pattern
because now the lines are
01:21:04.075 --> 01:21:05.200
really close to each other.
01:21:05.200 --> 01:21:09.846
The nodal line, we can
see clearly on the screen.
01:21:12.714 --> 01:21:17.270
Now I am going back down
to a lower frequency,
01:21:17.270 --> 01:21:20.660
and you can see that at
low frequency oscillation,
01:21:20.660 --> 01:21:24.650
the number of lines
is actually smaller,
01:21:24.650 --> 01:21:29.270
and that is because of
the smaller oscillation
01:21:29.270 --> 01:21:33.640
frequency and the
longer wavelengths
01:21:33.640 --> 01:21:36.090
of the normal modes.