Learners might have learned the basic concepts of the acoustics from the ‘Introduction to Acoustics (Part 1).’ Now it is time to apply to the real situation and develop their own acoustical application. Learners will analyze the radiation, scattering, and diffraction phenomenon with the Kirchhoff –Helmholtz Equation. Then learners will design their own reverberation room or ducts that fulfill the condition they have set up.
Lecture 1-1 Part 3. Breathing Sphere Problem 2
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From the course by Korea Advanced Institute of Science and Technology
Introduction to Acoustics (Part 2)
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From the lesson
RADIATION
You will be going to learn breathing sphere problem, then compare with the trembling sphere problem.
Meet the Instructors
Yang-Hann KimProfessor
Mechanical Engineering
And it look very complicated.
There is a ka, or there is an a square and a rho 0.
And there is an omega, so let's convert this omega using this relation.
The k equal to omega over c.
Therefore, I can write omega has to be equal to,
k times c, so let's put this as kc.
And we use this over there, then I have minus rho 0 c.
I have a characteristic impedance that
has a physical meaning, and I have k.
And then there is a a, so let's move k over here.
So there is ka multiply a.
So let me rearrange this,
then I can write over here minus rho 0 c.
And I eliminate this,
and I have -j still.
And I have this term.
So I will write over there -jka.
And I have this phase term over here still.
All right.
[COUGH] Okay,
then I can move this ka square as ka.
And then I use a over there.
So this term is interesting because it measures relative
distance compared with the radiator's size.
And the pressure depends on ka and the characteristic impedance.
And also it depends on U 0.
All that makes sense.
And this term is interesting.
If we see the magnitude of this term,
then it look like I have -1 and
plus jka, so magnitude of this term
strongly depends on ka squared.
So let me change this term.
And rather physically sensible term,
it turns out that I have, if I multiply -1,
-jk, but it is not convenient.
But I have -1 over here, so I would take this minus and then correct this.
Then I will multiply 1+jk in both side.
This will give me the denominator would be, what is it?
1-, if I multiply 1+jk,
I have 1+ka square.
And over here I have ka and
then I have 1+jka.
So the real part is ka divided by 1+(ka) square,
and the imaginary part is ka square divided by 1+(ka) square.
Okay, that is interesting.
The real part, of course there is another part, but
essential part that describes the physics associated
with the pressure is that the real part is (ka), 1+(ka) square, and
the imaginary part is (ka) square divided by 1+(ka) square.
So that is interesting.
The pressure depends on ka.
ka is a measure of radiation obviously.
And then ka means the size of a radiator compared with the wavelengths.
If ka is getting smaller and smaller,
Say ka is 0.1, then this denominator will approach to 1,
and it is dominated by (ka) square or ka.
So when ka is small, then I can argue
the real part is getting larger
compared with the imaginary part.
What it means?
What it means?
The real part is larger than imaginary part.
The real part is something related with the pressure and
real size, and this is imaginary part.
And that does not really provide us clear understanding about the figures
behind of the radiate.
Of course, you say, when ka is getting larger and larger,
this means that radiator size is large compared with the wavelength.
Then this term, because ka is getting larger and
larger, this term will approach to 1.
And this term approach to what?
ka is larger and larger, 1+(ka) square is approaching to (ka) square.
Therefore, it approach to 1 over ka, right?
So it certainly says many things, but not much.
What is the measure that can show us the radiation characteristics?
Pressure, velocity, radiation impedance.
So let's look at radiation impedance, Z r.
That is defined as the pressure in r direction,
And the velocity, For
given frequency.
Say I select omega.
This radiation impedance essentially shows how the ability
to radiate the sound into space with respect to r and P.
Okay.
So let's look at what it looks like by seeing the,
Okay, let's first see the velocity and pressure.
Okay, it is very interesting.
The reason why you have this term is because in the previous
derivation we have a exponential -jka, right?
Everybody remember?
That's why we included that one over here.
That is -jkr and minus, minus, plus, and there is a minus.
Therefore, this term is
reflected in this formation.
And of course this term,
this expression is valid when r is greater than, equal to a.
So this expression is good,
because I am seeing the phase away from
a because I subtract the a from r, r-a.
Okay, let's look at other term.
Okay, velocity is proportional to
the velocity on the surface magnitude U 0, that makes sense.
And the velocity is proportional to a over r square.
That is interesting.
So that means as we're far away from the source,
that means that r is getting larger and larger,
the velocity is decreasing proportional to 1 over r square.
Okay, that sounds good, because to conserve
the energy, which we will see later on,
as the surface is getting larger and larger,
Okay, area of surface is 4 pi i square.
So because the velocity is proportional to 1 over r square,
if you multiply the area certainly velocity through the,
the surface would be same if it is a 4 pi r square.
Okay, the pressure has the same form and interesting thing is this term.
Okay, this term.
So to see the physics, let's see the radiation impedance,
that is, the pressure divided by velocity.
And it look like that at the position r.
Now, This is the real term.
The real term shows how effectively radiate.
In other words, the real term simply shows that the fluid
particle is moving in phase with the pressure.
In other words, when I push the fluid with a pressure P,
the real parts says the motion of fluid in phase with a pressure.
That has the term look like that.
This is the imaginary part.
Thus the term reflect the fluid
particles moves with r of phase of j.
So that is a reactive term.
And interestingly, the numerator of real term is proportional to (kr) square,
and imaginary part of the numerator is proportional to kr.
Okay.
So when kr is much, much larger than 1, what it means physically?
Distance is r is very large compared with the lambda.
That means we are far away from the source.
That we call far-field.
In the far-field,
because kr is very large,
the real term approach to 1.
That means real term approach to rho 0 c.
That means in the far-field,
radiation impedance approach to the case of plane wave.
Therefore, in the far-field the wave propagate as if it is plane wave.
That make sense.
There's a breathing sphere in the far-field wave front looks planar,
therefore wave is propagating as a plane wave.
And let's look at the imaginary part.
When kr is very large, then what's going to happen?
0 again?
Okay, you said this is 0 and this is 0 too.
But it's a different function, right?
[COUGH] For rate of decreasing, or
if you look at the rate of decreasing of these
two term we look at, when kr is very large,
then this will approach to 1 and
this will approach to say kr is 1 million.
1 million square is much, much, much larger than 1 million.
Therefore, it approach to negligible value.
So in a far-field, because wave propagates as a plane wave,
the reactive term is very small.
So for breathing sphere in a far-field, in other words,
when k r is very large than 1,
then what you feel is almost the same as the plane wave field.
What about in a near-field, in other words,
when kr is much, much smaller than 1?
That means kr is simply, kr is lambda over r.
So when kr is small, that means
we are very close to the source, because r is small compared with lambda.
And in this case this term kr is very small.
Then 1+(kr) square is about 1.
Okay. And
it is approximately have us behavior of chaos care.
And this one, the imaginary part will
behave approximately as kr.
So one behave like kr scale and one behave like kr.
In a near field.
So in a near field as we can anticipate the behaviour
of wavefront is quite different compared with the behaviour of a far field.
That's why in the near field we can hear many different sound.
For example if you hear the sound very close to this radiator what
we hear would be somewhat different than what we hear far away.
That's the concept of near-field and far-field.
Okay, the other way to look at this property
would be seeing this impedance graphically.
Okay, so this shows the radiation impedance
normalized by Z with respect to kr first.
Okay, forget about this term in the meantime.
Okay, this is the real part.
As you can see, as kr getting large and large, this approach to 1.
As a plain wave.
But over here there's some interesting characteristics.
If you draw the imaginary part, it look like that.
In a far-field the contribution of an imaginary part is small.
Over here, the contribution of the real part and imaginary part is equally likely.
Okay.
So this area, we call near field.
There are some contribution from the real part of the radiation
impedance as well as the measurement part of the radiation impedance.
Okay.
That's pretty interesting, so
let's summarize what we learned today, okay.
I argued that understanding breathing sphere and
trembling sphere certainly make a foundation
to understanding to understand for
example general scattering radiation,
even diffraction.
And I demonstrate using Korean And
other musical instrument, in Korea we call it [FOREIGN]
And I should demonstrate how it radiates.
And the radiation depends on the angle.
Later on by studying the Sphere case we will understand
the interactivity later on.
So you experience the directivity pattern and
then I move on to the, to get the solution of mono,
a braiding sphere, by introducing with the velocity potential,
the reason why we use the velocity
potential is theoretically It is a simpler,
compared with just using pressure because the derivative of the lost potential
with respect to certain direction will give us the velocity in that direction.
And the pressure is simply.
So having one so one solution we can just use the derivative or
we can velocity as well as the pressure.
So that is convenient.
And then we are tempted to solve the tremblings of a breathings field case
I say prescribed the bountity of the loss in the surface
of radiant field is used 0 as potential of -0 without fee.
And then regard to solution using the loss of potential and
we try to see the physics in terms of radiation
impedance z r, which is quite similar with
the driving point impedance of a stream.
And you can, you might remember that the driving point impedance.
You remember that,
Driving point impedance of infinite stream
Z, m I think I used was row 0 CL,
that is speed of propagation of a string.
And the driving point is P is always 3, and
it has a war, and that look like what?
There's only imaginary part and a rogue 0C and
it has the form of co-tangents.
K l. Right, so driving point of a string
depends on the length and then k.
It also measure the length square with respect to lambda,
k Therefore the Point
impedance has only measured part purely reactive.
In this case it is purely Because it has only the real part
and the real part is the same as 0C, that is the characteristic impedance.
So for the gradient sphere case, in a far field,
rather than point impedance or regulation impedance in this case is equal to
0C and then the Of this part will move to 0 and
then in the mean time in the Field it has driving point in Or
radiation in And it has real part as well as imaginary part.
Because see here is a sphere.
Suppose you are a particle over here.
Okay, in the far field the average particle will move at the same
time right there.
Therefore, the radiation impedance in the far field will be same as the planar.
Over here, on a sphere, it has to move this way and that way.
There is some space which is not the same as the space that in a Field.
So there is a sum, finiteness,
or reactive action in the Field.
So let's move, let's keep talking about
the Spheres related to physics in the next lecture.
And I hope you can understand more in the next
lecture about the physics behind the breathing sphere.
Okay?
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