This course introduces acoustics by using the concept of impedance. The course starts with vibrations and waves, demonstrating how vibration can be envisaged as a kind of wave, mathematically and physically. They are realized by one-dimensional examples, which provide mathematically simplest but clear enough physical insights. Then the part 1 ends with explaining waves on a flat surface of discontinuity, demonstrating how propagation characteristics of waves change in space where there is a distributed impedance mismatch.
Lecture 3-2 Part 3. Solutions of the Wave Equation
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来自 Korea Advanced Institute of Science and Technology 的课程
Intro to Acoustics (Part 1)
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从本节课中
Acoustic Wave Equation and its Basic Physical Measures
On this week, you will learn deeply into acoustic intensity and energy. Then you will get the concept of units of sound, or dB scale with listening to various sound. At the end, you will be able to derive the solutions of the wave equation.
与讲师见面
Yang-Hann KimProfessor
Mechanical Engineering
Okay now as you know we have
a governing equation which can be written like that.
I will introduce three typical solutions.
Okay.
One typical solution will represent the plane wave.
One dimensional wave in access is.
Right.
So this is the wave propagating in this direction, right, going away.
Okay, if it is propagating in certain direction,
I mean in space, at certain production r.
Okay.
And then three dimensional form of wave can be expressed like that.
Those that have a same amount of t but I have k r.
What is k r?
K is wave number, okay that has three component.
If I express in x and y direction.
Okay.
So this is wave number vector, right.
So in this case I have wave propagation direction in this direction
So actually this is the projected
wave number on this surface, right?
As I showed here this is vector this is
the direction of wave number Okay.
If wave is propagating over there, then the projected wave,
the k dub r is simply how much is projected in this direction.
So this is quite generous three dimensional plane wave,
the reason why we called this plane wave is simply because wave front is plane.
Okay, wave front is plane.
The K essentially.
The direction normal to the plane.
Right?
It's very interesting.
And also that this is, because this is plane wave.
Okay?
If it can't collide the velocity
uxt using Euler's equation,
then we can obtain easily
the relation between velocity and
pressure for plane wave p over u.
Is zero that is row 0C.
So the velocity is simply 1/row
0c*pressure.
Okay?
So for the plane wave.
Impedance.
That is pressure divided by velocity zero c.
And that is the same as the characteristic impedance of.
That means there will be no reflection.
So everything is propagating because the pressure and velocity has in phase.
Okay now what if I have the surface of propagation or
in other words if I have this kind of wave.
That I want to see.
So let's investigate what's going on if I have the sound pressure look like that.
This is called monopole.
The reason why we call this a monopole is because 1/r behavior.
1/r means it has a singularity at r = 0.
At r = 0, the pressure has to go to infinity.
That's why we call this monopole, which means that we have one pole.
Pole simply means that at the pole we have an infinite barrier.
And using all the equation, we can find velocity like that.
How to find it, we take a derivative of pressure with respect to R,
put the minus and then, we divide this one by rho general.
And then we integrate and with respect to time, we will get this.
And the impedance.
The reason why we look at impedance is because
impedance exhibits the characteristics of wave propagation.
Remember, recall.
When we have infinite docks that does not have any reflection.
What we have?
The impedance.
Was rho 0C.
And the intensity, how does it look like?
Intensity, it fluctuates with the frequency of omega.
But that is mean intensity on that has a PC value, right.
But when we have a finite we have
a very reactive intensity right.
So certainly in this lecture I emphasized the intensity
as well as impedance as a major that exhibits
the characteristics of acoustic wave preparation.
Okay now, we, we discussed extensively about the meaning of
activity in test D and the reactors in test D.
But now lets go back to the concept of impedance, which again
exhibited the characteristic, what our characteristics are with propagation.
And for the monopole case, the impedance looks like that.
Okay this is the highlight.
It shows that it is a function of kr.
Okay row 0c is just the characteristic impedance of medium.
What does it mean?
Okay if k r is very large that means k r is simply 2 Pi.
Are divided by lambda, okay, because k equals to two pi over lambda.
If kr is very large, in other words,
the position I am looking at r is very.
Long compared with the wave lengths.
For instance, when I make a one kilohertz sound of monopole [SOUND].
The wave lengths is about 34 centimeter and the wave front that,
I mean impedance observed by those guy over there will be when kr is very large.
Then when kr is very large, this term goes to one, right?
That means the real part of the impedance approached to row
0 c That means when kr is very large,
impedance approach to as if the wave front propagates as a plane wave.
So in other words, when kr is large.
The wave front propagates as if it is a plane wave.
What about
the imaginary term over here?
When kr is large, sat kr is 100, the scale of 100 is 10,000.
10,000 plus 1 is 10,001.
One there's nothing and
I divide 100 by 10,000
that is ten to the minus two.
That is very small so
that means when k r is very large the imaginary term is negligible.
What it means?
That means, the imaginary part of impedance in only at the.
In other words, when kr is large, the impedance approach
to the plane wave case.
So that is interesting.
So let's see in more detail how it behaves.
So I normalized the impedance with the row 0c,
which means that if it approaches one, that means it is a plane wave.
It behaves as a plane wave.
Okay so this graph is very interesting.
Let me use okay about over here.
So which is the case kr is more than like a seven or
eight, we can assume this area,
I mean this field is plane wave and we call
We call this case far field.
Far field, far with respect to wavelength.
I emphasize again kr is the measure of
the distance with respect to wavelength.
All right that's interesting.
In the view field this is I'll show this area is a near field.
Okay.
Near field.
As you can see, this is the function
(kr) squared / 1 + (kr) squared.
And this is the function kr / 1 + (kr) squared.
That this represents the imaginary part of the impedance as you can see over here
the imaginary part of impedance approach to zero in a far field.
There is no reactive term in the far field.
But in this field this is the close rule of the area over here.
And we can see that most likely there is
equally likely participation of real part as well as imaginary part.
In other words in a near field.
Okay when you approach when I make one kilohertz sound.
[SOUND] In this field the impedance has
real power as well as imaginary power.
That means you cannot assume the the wave
propagation over here is plane wave.
There is almost equally likely participation
of the real part, and imaginary part.
This part shows the pressure and
particle velocity in the near field.
This is magnitude.
And this is a phase.
And as this graph shows,
the magnitude of pressure and phase of pressure.
As we can see here, in a far field the pressure magnitude is almost constant.
And the phase shows that it propagates as a spherical wave.
Over here, near field, magnitude depends on the position.
Okay.
Phase, there is quite different to what we can see in the far field.
All right. That's interesting.
Okay, this is the case of dipole.
Okay, dipole is the sound wave and
I locate two monopole very close nearby.
And what will happen?
The typical dipole of sound could be something generated by
the drum for example because drum oscillate like that.
Therefore, there is a sound radiating in this two direction and
the small radiation in this direction.
That is dipole, so sound directivity of dipole.
Look like a radiate sound like that.
Okay.
And looking at the impedance of dipole it looked like that is more complicated.
But still the impedance in a far field.
Of dipole is quite similar with the monopole but
reactive part of dipole became more slow.
Than the monopole case, as you can see here the phase of
dipole in far field it look like that compare with what previously you saw,
all right.
This is the case of quadrapole, okay, in far field, very similar behavior.
Okay, I'd like to summarize chapter two.
We have attempted to understand how acoustic waves are generated and
propagated in a compressible fluid.
By using conservation of mass and state equation.
And the Newton second law for unit surface and unit volume.
And we found that acoustic intensity expresses the direction of acoustic
power flow as well as its magnitude.
And the intensity certainly expressed how the acoustic energy,
Is balanced by.
Intensely flux through the surface.
And we studied a way to measure the associated acoustic variables
in accordance with human perception.
We introduced the db scale and the dba.
And also, we introduced the octave scale.
And we had some demonstration.
And we also have the possible solution.
We didn't study this, possible solution method, we didn't study this.
We will do that later on.
[INAUDIBLE] And also we studied.
About the monopole, dipole,
quadrapole characteristics of sound.
Okay?
That mostly covers what I intend to talk to you guys today.
