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## The Doppler effect

# When the source and the wave move at the same velocity

## Video transcript

In the last several videos, we
assumed that the velocity of the source, of the object that's
emitting the wave, we've assumed that that's less
than the velocity of the wave. And we saw what happens
with the Doppler effect and all of that. But what I want to do in this
video is not make this assumption. In particular, let's see what
happens, at least, at first for our formulas, and then get
a conceptual understanding. Let's see what happens when the
velocity of the source is equal to the velocity
of the wave. The first thing we might try to
do is just apply this new assumption into the formulas
that we had in the last video. And those formulas were here. These are the observed period
and frequency for an observer that's in the direction
of the object. And if we make this assumption,
that the velocity of the sound-- and the velocity
of the source-- we're not necessarily dealing with
soundwaves, although that might be an easy visualization
for you. That tends to be how
I visualize it. But what happens to these
formulas when the velocity of the source is equal to the
velocity of the wave? If these two quantities are
equal up here, you have something, subtract the
same thing from it. This numerator right
here becomes 0. So it'll turn this whole
thing into being 0. So the period, or the observed
period, will be 0, which means you don't have to wait
any time at all between successive crests. The entire waveform just gets
infinitely bunched together. So it's kind of like
one impulse. And if we look at the frequency,
we can either look directly at the formula, and
you'll see, you have something divided by 0 right here. So you can say this is 1 over 0,
or you could just say that the frequency is 1 over the
period, and you get this thing that's undefined. But if you want to think about,
what does the frequency approach as the velocity of
the source approaches the velocity of the wave, if this
thing is only a little bit less that thing, this is going
to be a very, very, very small, very, very small
positive number. So when you divide these
quantities by that very, very, very small positive
number, you're going to approach infinity. So the frequency is undefined at
the speed, at the velocity of the wave, but it's going
to approach infinity. It will approach infinity as
the source approaches the velocity of the wave. Not
necessarily a soundwave. I keep using soundwaves, because
that's how I tend to visualize things. And we'll talk in future videos
specifically about soundwaves. And we'll touch on it a little
bit in this video. So what is this telling us? Does this make any sense? And if you think about it, at
least to me, it starts to make sense just what you saw in the
last couple of videos. The last couple of videos,
when something was moving slower than the speed
of sound, you had, OK, I'm here now. And I'm about to release the
next crest. If I go one period ago, maybe I was right there. And the crest that I had
released at that time period maybe has traveled this
far, just like that. If you go a period before
that, I would have been over there. And crest that I released then
would have traveled that far. We saw this in the
last two videos. And if you go the period
before that, I would have been there. And the crest that I would
have released would have been that far. This was the whole reason why
the Doppler effect happens. Because the observer sitting
right here-- let me do this in a separate color-- the observer
sitting right here is going to experience these crests
more frequently than an observer sitting out here. Because the wavelength gets
compressed, because every time this guy releases a new
crest, or a new cycle, he has moved forward. He's moved forward in the
direction of this motion right here. So let's think about what
happens when he is exactly moving at the speed of the wave.
So let's say that the source is here now. This is right where he is. And he's right about to release
a new crest. So where was he at one period ago? So let's say he was here
one period ago. So one period ago, if he's right
going to release crest or cycle right now,
one period ago, he released another cycle. And where has that cycle gone? Well, we're assuming that the
wave is traveling at the same velocity as this guy. But it's going radially
outward. So whatever he released then,
it will have traveled at the same velocity as himself. So it will have gotten
this far. He released it one period ago,
and that's where he was one period ago. Over the course of the next
period, he traveled there and so did the wave. The wave
also traveled there. Now, where was this character
two periods ago? When I talk about the period,
I'm talking about the actual period of the wave. Every
period, or how long does it take between similar points
in the cycle? And I like to think of them as
the crests in the cycle. So two periods ago,
he was here. And he released a-- you can
imagine, a pulse, or crest. And where will that be now? Well that will have traveled
as far as he did. He traveled that far,
and so will the pulse that he released. It will have traveled-- actually
I have to make it a little bit more symmetric. It will have traveled
that far. And if you go three periods ago,
I think you get the idea. If you go three periods
ago, he was here. And he released a pulse then,
or crest, or a cycle of the wave. And where will
that be now? Well, it will have traveled
as fast as he's gotten. So it will have gotten
this far. Of course, it's traveling that
velocity in every direction, radially outward. Now think about the situation
for the observer. Think, in particular, about
the observer who's sitting right here. Let's say he's just out of the
way so that this thing doesn't run into him and kill him,
and what whatever else. But he's just out of the way,
just enough to experience the sound, but not directly collide
with this object that's emitting the-- well, I
shouldn't say sound-- emitting the wave. I want to be
general right here. We're not assuming that
this is a soundwave. What's he going to experience? Well, he's not going to-- well,
if we assume this is a soundwave, he's not going to
hear anything until the thing passes right there. And right when the thing passes,
it has all of the sound that it generated coming
with it at exactly that moment in time. Instead of hearing things
periodically, all of the wave fronts smack this
guy all at once. And perceptually, instead of
hearing a pitch, because you're hearing something
periodic, you're just going to hear a big thump, because all
of that sound energy is getting to you at
the same time. You're just going
to hear a thump. Because it's no longer
really a frequency. All of the energy is coming
to you at the same time. And when you are talking about
sound, specifically, and especially, when you are
trans-sonic, which means you're around the speed of
sound, or parts of you are above or below the speed of
sound, and you move into supersonic speeds, that's
what people relate to the sonic boom. And we're going to talk a little
bit more about that in the next video and mach numbers,
all that, because I just find all of that
fascinating. But I think this is intuitive. Because when you look at this,
everything is just reaching you at exactly the same time. And this was the case of
soundwaves, but it would be true of any type of waves.