Now what happens if the
source moves really quickly?
What happens when the
source moves faster
than the speed of the
waves in that medium?
So imagine that we have
this supersonic jet here,
travelling at some speed v
subscript s-- s is for source,
again--
which is faster than the
speed of sound in that medium.
We'll have a look
at a simulation,
and then we'll do the maths
behind it to show what happens.
We actually end up with
getting a shockwave.
So this animation
shows what happens
when a object moves faster than
the wave speed in that medium.
You can see you get
pressure peaks forming.
This shows the same thing
in three dimensions.
So we saw in the animation that
we get these pressure waves
travelling along where all
the wavefronts are together.
So what we're actually
interested in is what's
this angle theta, which
this wavefront makes
with this horizontal line here?
So to work this out, let's
start with the our plane
at some point here, at
some initial time t0.
Now at some time
t, This shows where
the wavefront has gotten to.
So the distance between here
and here is given by vt.
That's how fast this wave
travels through the medium.
Now in the same time, the jet
has actually travelled a bit
further than that.
It's travelled some
distance here, vst.
Now you can see at that point,
the jet is going to let out a--
it's going to continue to
let out these soundwaves.
So we'll have the smaller
circles centred on its path
as it travels along.
So we have a series
of circles like this.
And so if we want to work
out the angle theta here,
we actually draw a
tangent from the point
where the plane is up
to this point here.
And that angle in
there is theta.
And now we can use
simple geometry.
Because this is a radius,
and this is a tangent,
the angle between those
two are 90 degrees.
So we can just use right
angle triangle geometry.
And so we know that sin
theta is equal to opposite
over hypotenuse.
And so sin theta is equal
to opposite vt over vst.
And you can see the
t's cancels out.
So that's equal to v
over vs. So we've now
come up with an expression
for sin theta, the angle
that our shockwave makes with
the path of the moving object,
the source of the sound.
So this animation shows the
shockwave hitting the ground.
When that wavefront hits you, if
you're standing on the ground,
you hear what's known
as a sonic boom, which
is why when there's a supersonic
aircraft going overhead,
you hear that boom noise.
Now this is a very
famous photograph
showing a supersonic
aircraft just
hitting that sound barrier.
It causes the water to
condense around it, which
is what you're seeing here.
OK.
Now we've had a
look at this angle
here and how to derive it.
But one thing that you've
probably heard of before
is Mach numbers.
So the Mach number is
actually the inverse
of this sin theta relationship.
It's vs, the velocity of
the source, divided by v.
And as we've just said,
this conical wavefront here
is known as a shockwave.
Now you may be
surprised to know--
but you've probably actually
seen shockwaves before--
when a boat goes
faster than the speed
that the waves travel through
the water, you get a shockwave.
So with this fast boat
here, this speed boat,
you can see the
shockwave behind it.
This photo here shows a bullet.
And you can see the shockwaves
generated in the air
as the bullet is going faster
than the speed of sound in air.
OK, so a question for you
to do-- an airplane flying
with a constant velocity
moves from a cold air mass
into a warm air mass.
Does the Mach number increase,
decrease, or stay the same?
