0:00:00.870,0:00:04.170
I've got this source of a wave[br]right here that's moving to

0:00:04.170,0:00:06.150
the right at some velocity.

0:00:06.150,0:00:14.510
So let's just say that the[br]velocity of the source-- let's

0:00:14.510,0:00:19.460
call it v sub s to the right--[br]so we're really going to do

0:00:19.460,0:00:21.180
what we do in the last video,[br]but we're going to do it in

0:00:21.180,0:00:23.630
more abstract terms so we can[br]come up with a generalized

0:00:23.630,0:00:26.160
formula for the observed[br]frequency.

0:00:26.160,0:00:30.190
So that's how fast he's moving[br]to the right, and he's

0:00:30.190,0:00:35.400
emitting a wave. Let's say the[br]wave that he's emitting-- so

0:00:35.400,0:00:40.930
the velocity of wave--[br]let's call that v

0:00:40.930,0:00:43.270
sub w radially outward.

0:00:43.270,0:00:45.510
We've got to give a magnitude[br]and a direction.

0:00:45.510,0:00:46.760
So radially outward.

0:00:50.070,0:00:53.660
That's the velocity of the wave,[br]and that wave is going

0:00:53.660,0:00:55.950
to have a period and a[br]frequency, but it's going to

0:00:55.950,0:00:58.550
have a period and a frequency[br]associated from the point of

0:00:58.550,0:01:00.630
view of the source.

0:01:00.630,0:01:01.515
And we're going to[br]do everything.

0:01:01.515,0:01:02.840
This is all classical[br]mechanics.

0:01:02.840,0:01:04.959
We're not going to be talking[br]about relativistic speed, so

0:01:04.959,0:01:07.580
we don't have to worry about all[br]of the strange things that

0:01:07.580,0:01:10.580
happen as things approach[br]the speed of light.

0:01:10.580,0:01:15.670
So let's just say it has[br]some period of-- let me

0:01:15.670,0:01:17.130
write it this way.

0:01:17.130,0:01:22.220
The source period, which is the[br]period of the wave from

0:01:22.220,0:01:26.890
the perspective of the source,[br]so the source period, we'll

0:01:26.890,0:01:31.850
call it t sub source, And the[br]source frequency, which would

0:01:31.850,0:01:34.400
just be-- we've learned,[br]hopefully it's intuitive now--

0:01:34.400,0:01:35.900
would be the inverse of this.

0:01:35.900,0:01:42.790
So the source frequency would[br]be-- we'll call it f sub s.

0:01:42.790,0:01:44.960
And these two things are the[br]inverse of each other.

0:01:44.960,0:01:47.300
The inverse of the period[br]of a wave is its

0:01:47.300,0:01:49.060
frequency, vice versa.

0:01:49.060,0:01:52.010
So let's think about what's[br]going to happen.

0:01:52.010,0:01:56.930
Let's say at time equal zero,[br]he emits that first crest,

0:01:56.930,0:01:59.430
that first pulse, so he's[br]just emitted it.

0:01:59.430,0:02:01.540
You can't even see it because[br]it just got emitted.

0:02:01.540,0:02:05.020
And now let's fast forward[br]t seconds.

0:02:05.020,0:02:07.695
Let's say that this is in[br]seconds, so every t seconds,

0:02:07.695,0:02:09.410
it emits a new pulse.

0:02:09.410,0:02:11.970
First of all, where is[br]that first pulse

0:02:11.970,0:02:14.800
after t sub s seconds?

0:02:14.800,0:02:18.170
Well, you multiply the velocity[br]of that first pulse

0:02:18.170,0:02:19.650
times the time.

0:02:19.650,0:02:22.780
Velocity times time is going[br]to give you a distance.

0:02:22.780,0:02:24.200
If you don't believe me, I'll[br]show you an example.

0:02:24.200,0:02:27.980
If I tell you the velocity is 5[br]meters per second, and let's

0:02:27.980,0:02:30.850
say that this period is 2[br]seconds, that's going to give

0:02:30.850,0:02:32.340
you 10 meters.

0:02:32.340,0:02:34.580
The seconds cancel out.

0:02:34.580,0:02:38.230
So to figure out how far that[br]wave will have gone after t

0:02:38.230,0:02:42.090
sub s seconds, you just multiply[br]t sub s times the

0:02:42.090,0:02:43.830
velocity of the wave.

0:02:43.830,0:02:46.160
And let's say it's[br]gotten over here.

0:02:46.160,0:02:47.960
It's radially outward.

0:02:47.960,0:02:50.080
So, I'll draw it radially[br]outward.

0:02:50.080,0:02:53.690
That's my best attempt[br]at a circle.

0:02:53.690,0:03:00.080
And this distance right here,[br]this radius right there, that

0:03:00.080,0:03:03.140
is equal to velocity[br]times time.

0:03:03.140,0:03:08.780
The velocity of that first[br]pulse, v sub w, that's

0:03:08.780,0:03:09.510
actually the speed.

0:03:09.510,0:03:11.630
I'm saying it's v sub[br]w radially outward.

0:03:11.630,0:03:12.800
This isn't a vector quantity.

0:03:12.800,0:03:14.750
This is just a number[br]you can imagine.

0:03:14.750,0:03:19.450
v sub w times the period,[br]times t of s.

0:03:22.100,0:03:23.780
I know it's abstract, but just[br]think, this is just the

0:03:23.780,0:03:25.300
distance times the time.

0:03:25.300,0:03:29.330
If this was moving at 10 meters[br]per second and if the

0:03:29.330,0:03:31.190
period is 2 seconds,[br]this is how far.

0:03:31.190,0:03:34.520
It will have gone 10 meters[br]after 2 seconds.

0:03:34.520,0:03:36.635
Now, this thing we said[br]at the beginning of

0:03:36.635,0:03:38.100
the video is moving.

0:03:38.100,0:03:40.410
So although this is radially[br]outward from the point at

0:03:40.410,0:03:43.340
which it was emitted, this thing[br]isn't standing still.

0:03:43.340,0:03:44.720
We saw this in the last video.

0:03:44.720,0:03:46.690
This thing has also moved.

0:03:46.690,0:03:47.540
How far?

0:03:47.540,0:03:48.700
Well, we do the same thing.

0:03:48.700,0:03:52.330
We multiply its velocity times[br]the same number of time.

0:03:52.330,0:03:55.770
Remember, we're saying what does[br]this look like after t

0:03:55.770,0:03:59.230
sub s seconds, or some period[br]of time t sub s.

0:03:59.230,0:04:01.130
Well, this thing is moving[br]to the right.

0:04:01.130,0:04:02.660
Let's say it's here.

0:04:02.660,0:04:05.570
Let's say it's moved[br]right over here.

0:04:05.570,0:04:08.470
In this video, we're assuming[br]that the velocity of our

0:04:08.470,0:04:12.440
source is strictly less than the[br]velocity of the wave. Some

0:04:12.440,0:04:14.750
pretty interesting things happen[br]right when they're

0:04:14.750,0:04:16.890
equal, and, obviously, when[br]it goes the other way.

0:04:16.890,0:04:18.860
But we're going to assume that[br]it's strictly less than.

0:04:18.860,0:04:23.150
The source is traveling slower[br]than the actual wave.

0:04:23.150,0:04:24.290
But what is this distance?

0:04:24.290,0:04:25.980
Remember, we're talking[br]about-- let me do it

0:04:25.980,0:04:27.550
in orange as well.

0:04:27.550,0:04:31.650
This orange reality is what's[br]happened after t sub s

0:04:31.650,0:04:33.160
seconds, you can say.

0:04:33.160,0:04:35.360
So this distance right here.

0:04:35.360,0:04:38.390
That distance right there--[br]I'll do it in a different

0:04:38.390,0:04:41.540
color-- is going to be the[br]velocity of the source.

0:04:41.540,0:04:46.350
It's going to be v sub[br]s times the amount of

0:04:46.350,0:04:47.190
time that's gone by.

0:04:47.190,0:04:49.340
And I said at the beginning,[br]that amount of time is the

0:04:49.340,0:04:51.480
period of the wave. That's[br]the time in question.

0:04:51.480,0:04:54.340
So period of the wave t sub s.

0:04:54.340,0:04:57.950
So after one period of the wave,[br]if that's 5 seconds,

0:04:57.950,0:05:00.910
then we'll say, after 5 seconds,[br]the source has moved

0:05:00.910,0:05:07.800
this far, v sub s times t sub s,[br]and that first crest of our

0:05:07.800,0:05:12.300
wave has moved that far,[br]V sub w times t sub s.

0:05:12.300,0:05:14.300
Now, the time that we're talking[br]about, that's the

0:05:14.300,0:05:16.270
period of the wave[br]being emitted.

0:05:16.270,0:05:19.810
So exactly after that amount of[br]time, this guy is ready to

0:05:19.810,0:05:21.960
emit the next crest.[br]He has gone

0:05:21.960,0:05:23.500
through exactly one cycle.

0:05:23.500,0:05:27.390
So he is going to emit[br]something right now.

0:05:27.390,0:05:30.680
So it's just getting emitted[br]right at that point.

0:05:30.680,0:05:33.760
So what is the distance between[br]the crest that he

0:05:33.760,0:05:37.990
emitted t sub s seconds ago or[br]hours ago or microseconds ago,

0:05:37.990,0:05:38.730
we don't know.

0:05:38.730,0:05:41.650
What's the distance between this[br]crest and the one that

0:05:41.650,0:05:43.210
he's just emitting?

0:05:43.210,0:05:45.600
Well, they're going to move at[br]the same velocity, but this

0:05:45.600,0:05:48.810
guy is already out here, while[br]this guy is starting off from

0:05:48.810,0:05:50.260
the source's position.

0:05:50.260,0:05:52.530
So the difference in their[br]distance, at least when you

0:05:52.530,0:05:54.630
look at it this way, is the[br]distance between the source

0:05:54.630,0:05:56.670
here and this crest.

0:05:56.670,0:05:59.740
So what is this distance[br]right here?

0:05:59.740,0:06:02.790
What is that distance[br]right there?

0:06:02.790,0:06:07.270
Well, this whole radial[br]distance, we already said,

0:06:07.270,0:06:12.090
this whole radial distance is[br]v sub w, the velocity of the

0:06:12.090,0:06:16.440
wave, times the period of the[br]wave from the perspective of

0:06:16.440,0:06:19.170
the source, and we're going to[br]subtract out how far the

0:06:19.170,0:06:20.540
source itself has moved.

0:06:20.540,0:06:23.350
The source has moved in the[br]direction, in this case, if

0:06:23.350,0:06:24.910
we're looking at it from[br]this point of view,

0:06:24.910,0:06:27.050
of that wave front.

0:06:27.050,0:06:33.140
So it's going to be minus v[br]sub s, the velocity of the

0:06:33.140,0:06:38.990
source, times the period of the[br]wave from the perspective

0:06:38.990,0:06:40.340
of the source.

0:06:40.340,0:06:41.780
So let me ask you a question.

0:06:41.780,0:06:44.900
If you're sitting right here, if[br]you're the observer, you're

0:06:44.900,0:06:49.730
this guy right here, you're[br]sitting right over there, and

0:06:49.730,0:06:52.680
you've just had that first[br]crest, at that exact moment

0:06:52.680,0:06:56.190
that first crest has passed you[br]by, how long are you going

0:06:56.190,0:06:58.720
to have to wait for[br]the next crest?

0:06:58.720,0:07:01.520
How long until this one that[br]this guy's emitting right now

0:07:01.520,0:07:03.010
is going to pass you by?

0:07:03.010,0:07:04.920
Well, it's going to have[br]to cover this distance.

0:07:04.920,0:07:06.750
It's going to have to[br]cover that distance.

0:07:06.750,0:07:07.650
Let me write this down.

0:07:07.650,0:07:10.580
So the question I'm asking is[br]what is the period from the

0:07:10.580,0:07:14.060
point of view of this observer[br]that's right in the direction

0:07:14.060,0:07:15.320
of the movement of the source?

0:07:15.320,0:07:19.660
So the period from the point[br]of view of the observer is

0:07:19.660,0:07:22.180
going to be equal to the[br]distance that the next pulse

0:07:22.180,0:07:25.240
has to travel, which is that[br]business up there.

0:07:25.240,0:07:26.900
So let me copy and paste that.

0:07:30.260,0:07:32.390
So it's going to be that.

0:07:32.390,0:07:33.150
Let me get rid of that.

0:07:33.150,0:07:35.690
It shouldn't look like an equal[br]sign, so I can delete

0:07:35.690,0:07:36.540
that right over there.

0:07:36.540,0:07:39.070
Or a negative sign.

0:07:39.070,0:07:41.335
So it's going to be this[br]distance that the next pulse

0:07:41.335,0:07:42.840
is going to travel, that one[br]that's going to be emitted

0:07:42.840,0:07:46.260
right at that moment, divided by[br]the speed of that pulse, or

0:07:46.260,0:07:49.310
the speed of the wave, or the[br]velocity the wave, and we know

0:07:49.310,0:07:50.400
what that is.

0:07:50.400,0:07:52.565
That is v sub w.

0:07:58.680,0:08:01.600
Now this gives us the period[br]of the observation.

0:08:01.600,0:08:01.770
Now.

0:08:01.770,0:08:03.850
If we wanted the frequency-- and[br]we can manipulate this a

0:08:03.850,0:08:04.190
little bit.

0:08:04.190,0:08:05.980
Let's do that a little bit.

0:08:05.980,0:08:08.940
So we can also write this.

0:08:08.940,0:08:12.390
We could factor out the[br]period of the source.

0:08:12.390,0:08:14.630
So t sub s we could[br]factor out.

0:08:14.630,0:08:20.440
So it becomes t sub s times the[br]velocity of the wave minus

0:08:20.440,0:08:26.690
the velocity of the source, all[br]of that over the velocity

0:08:26.690,0:08:30.230
of the wave. And so just like[br]that, we've gotten our formula

0:08:30.230,0:08:33.308
for the observed period for this[br]observer who's sitting

0:08:33.308,0:08:38.270
right in the path of this moving[br]object as a function of

0:08:38.270,0:08:42.270
the actual period of this wave[br]source, the wave's velocity

0:08:42.270,0:08:44.930
and the velocity[br]of the source.

0:08:44.930,0:08:46.670
Now, if we wanted the frequency,[br]we just take the

0:08:46.670,0:08:48.140
inverse of this.

0:08:48.140,0:08:49.340
So let's do that.

0:08:49.340,0:08:52.640
So the frequency of the[br]observer-- so this is how many

0:08:52.640,0:08:54.630
seconds it takes for him[br]to see the next cycle.

0:08:54.630,0:08:57.110
If you want cycles per second,[br]you take the inverse.

0:08:57.110,0:08:58.880
So the frequency of the observer[br]is just going to be

0:08:58.880,0:08:59.600
the inverse of this.

0:08:59.600,0:09:02.060
So if we take the inverse of[br]this whole expression, we're

0:09:02.060,0:09:07.800
going to get 1 over t sub[br]s times v sub w over the

0:09:07.800,0:09:10.840
velocity of the wave minus[br]the velocity the source.

0:09:10.840,0:09:13.740
And of course, 1 over the period[br]from the point of view

0:09:13.740,0:09:16.580
of the source, this[br]is the same thing.

0:09:16.580,0:09:18.840
This right here is the[br]same thing as the

0:09:18.840,0:09:20.260
frequency of the source.

0:09:20.260,0:09:21.070
So there you have it.

0:09:21.070,0:09:22.160
We have our two relations.

0:09:22.160,0:09:26.010
At least if you are in the path,[br]if the velocity of the

0:09:26.010,0:09:28.690
source is going in[br]your direction,

0:09:28.690,0:09:30.310
then we have our formulas.

0:09:30.310,0:09:34.190
And I'll rewrite them, just[br]because the observed period of

0:09:34.190,0:09:37.600
the observer is going to be the[br]period from the point of

0:09:37.600,0:09:42.060
view of the source times the[br]velocity of the wave minus the

0:09:42.060,0:09:44.080
velocity of the source-- that's[br]the velocity of the

0:09:44.080,0:09:47.650
source-- divided by the velocity[br]of the wave itself.

0:09:47.650,0:09:51.640
The frequency, from the point[br]of view of this observer, is

0:09:51.640,0:09:53.560
just the inverse of that,[br]which is the frequency.

0:09:53.560,0:09:56.590
The inverse of the period is the[br]frequency from the point

0:09:56.590,0:10:00.840
of view of the source times[br]the velocity of the wave

0:10:00.840,0:10:03.730
divided by the velocity[br]of the wave minus the

0:10:03.730,0:10:05.450
velocity of the source.

0:10:05.450,0:10:07.640
In the next video, I'll do the[br]exact same exercise, but I'll

0:10:07.640,0:10:10.400
just think about what happens to[br]the observer that's sitting

0:10:10.400,0:10:11.960
right there.