Tip:
Highlight text to annotate it
X
Next up, we have the doppler effect
You have probably heard about the word "doppler" in other parts of your life, but here
very specifically, the doppler effect applies to waves in the sense that we are talking about how the perceived frequency changes
due to either the source or the observer in motion, or moving
it happens for all kinds of waves, of course, it happens for water wave, for instance, ,if you have
the wake that a boat makes as it moves through the water
or it could have to do with light waves which might be the red shift that you may have heard of
for us to infer that stars are moving away from us
and then also we have for sound, because we have been talking about sound so much
and in terms of sound, you are going to be changing pitches, because you frequency changes
one common example is, if you happen to be at a formula 1 race track
and then you are seeing this car coming at you really really fast
as it comes towards you, you are going to hear a high pitch whizzing [poor whizzing imitation]
and then as it moves pass you, it suddenly drops and the frequency would be lower [poor imitation of F1 car sound]
So qualitatively, let's take a look at how this works
with the help of the wonderful contributors at wikipedia
This contributor has illustrated
first of all, a stationary source, everything is symmetrical and nothing's moving
so this is something we are very familar with, we have a source in the middle emitting circular wavefronts
2D, could be spherical
and what these blue lines are, they are wavefronts, they are basically the crest of the wave
so the space between two adjacent lines, between these two blue lines, would be the wavelength of the wave
so right now nothing is moving, so it is all symmetrical, no matter where you stand you are going to hear the same sound
now we introduce motion
the source here is moving towards the right
and because it is moving, it is catching up with its own wavefront as it emits them
so in front of it you see, on this patch here, all the wavefronts get squished and closer together
because the source is catching up to them. Shorter wavelength means higher frequency
and then in the back here
the wavelengths get stretched out. Higher wavelength, therefore lower frequency
Important to note here, the wave itself is not moving any faster. It's the same medium
so then the phase speed is still the same
it's the fact that the source is moving that the wavelengths get crunched up so therefore
dividing through by the same v, you get a different frequency
but before we look at this quantitatively, let's quickly look at what happens
if you move faster than the speed of sound
so this is still the case when moving slower than the speed of sound
but there is nothing stopping you from moving faster than the speed of sound, and that's when this happens
if you are moving so fast that you are passing your own wavefront, what you leave behind is this cone of
all these different pressure adding up
you see this cone back here, that's what we call a sonic boom
sonic boom is not like one instant burst of noise as you break the sound barrier
as long as you are moving faster than the speed of sound, behind you, you are creating this big wall of pressure
continuously
it just sounds like it is a single boom because
this wavefront of really high pressure passes you, so you only hear it for a very short time, but it keeps moving
but you are constantly creating this high volume pressure wave behind you
moving onto the quantitative stuff
There are actually 2 cases to consider here. First either the source moving or the observer moving
and the derivation is a little different, so let's consider first the source moving
because the source is moving towards the right, we have the wavefronts compressed
here to the rights and then stretched out in behind
to keep things simple, we are going to consider the one dimensional case
where the source and the observer, the green dot here
is along the line of the motion so
everything is nice and in 1D
if there was 2D, we can take components and it still works, but we will keep it a little simpler
like that
so first off, we have to consider
what is this new wavelength
so we have the observer seeing this wavefront
the leading wavefront, and the wavefront behind it
would have been back here and that would have been the original wavelength
but
because the source has moved by the time it emits the second wave, it's moved forward
so it's actually going to sit a little out here, and that's going to be our new wavelength. Well, how long is this?
well, how fast it travels times some time, and the time of course is the period
because the period is the time between emitting 2 separate wavefronts
call that T_o
so
...
...
...
...
...
...
...
rearranging
...
...
...
v_s of course is the velocity of the source
and then this is the phase speed of the wave
so this is for the case where
the source is moving. Don't copy this down yet, we still have to complete the picture with
the observer moving as well
of course, this can be plus or minus depending on if you are moving towards or if you are moving away
and we will reason that out in a second, but that's how you derive the formula with the source moving
so the second case to consider is if the observer is moving instead of the source
because the source is staying the same, you notice all these wavefronts out here
they are all nice and symmetric, evenly spaced, and their wavefront is still separated by the oringal wavelength
because the sources is not moving. It is still eminating outward at a phase velocity v
but this time, the observer is moving, so the observer is going to hits subsequent wavefronts sooner than otherwise
with respect to the observer then
it's seeing that the wavefront is moving toward him relative to the observer to be
a new speed ...
... together
this new v is going to be the new frequency times the original wavelength because the wavelength hasn't changed
and of course the wavelength is the oringal phase velocity divided by the original frequency
so we can shuffle things around and you can see that the v_observe
the observer velocity ends up on top
while the source ends up on the bottom. Again, this can be plus or minus depending if you are moving towards or away
and here's the relationship when the observer is moving and how the frequency changes
quick aside: for some reason
this doesn't happen for light waves
the speed of light is the speed of light regardless of how fast the observer is moving
so that's where all this fun of special relativity comes from, but that's for another course
here, we are going to combine the
two formulae that we have for doppler
we have
...
...
and the two observer and the source parts play together the same way
combines together very well
the observer goes on top
the source goes on the bottom
and the plus or minus indicates whether you're moving towards or away. Now, we have to reason it out
the easiest way to remember this because the sign convention is different for the top and bottom, the easiest way to work it out is
to think: if I am the observer and is moving towards my source, am I going to get a higher or lower frequency
it should be higher, therefore I must use a plus on top to make it bigger
if I have a source that is moving away from me, for instance
would the frequency be higher or lower
the frequency will be lower
so I'll use plus on the bottom
to make the final frequency lower and this is how you would reason out whether to use plus or minus
but there you go, that is the equation
that we need for the doppler effect
now, for some examples