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 5-2 Part 3. Transmission and Reflection of a Finite Structure
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來自 Korea Advanced Institute of Science and Technology 的課程
Intro to Acoustics (Part 1)
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從本節課中
Waves on a Flat Surface of Discontinuity 2
You will going to learn more about the waves on a discontinuity. With the Snell's Law, you will be able to find out the wave propagation after transmission and reflection on discontinuity.
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Yang-Hann KimProfessor
Mechanical Engineering
So this summarize what
I wanted to argue today.
For case when we have instant wave then because
of the discontinuity we have reflected and PT.
And we prove that, this is equivalent with having
two PI which we called blocked pressure and
the radiation due to the vibration of which will vibrate minus.
Time so there will be some radiation.
This is PT.
So it's physically very straightforward.
If you have efficient radiation, you have many, I mean big sound pressure outside.
Therefore you have large PT.
For the noise control purposes, You would like to have a small PT.
In other words, no radiation.
No radiation is the case when you have rigid wall and that is not practical or
dismissable.
You might want to have some water that has some filtered characteristics.
In other words, what I meant by filtered is it has a vat and
less radiation efficiency among certain frequency band.
But has a good radiation efficiency in other band.
Then you can use that band to control the noise that you want to control.
So consider the frequency characteristics of this partition.
You may design good partition
by seeing the radiation of this partian.
I used the same concept to argue that
transmission loss of infinite partition or
infinite plate is smaller than transmission
coefficient of a finite plate.
By introducing this radiation constant, and then we will use
this radiation concept to explain later in the next chapter.
Radiation scatter and defraction in unified order.
So if you understand radiation you understand scattering,
defraction, everything.
So radiation is sort of key concept
that explains almost everything,
except the room acoustics, in acoustics.
So, LM4 is simply is a two blocked pressure and radiation.
Okay, if you have a more general partition that has spraying and damping.
Using this concept,
understand thing what's going to happen due to this kind of partition.
It's rather simple because there is a same blocked pressure and
then radiation of this partition that has not
only mass but also spray and.
Mass law is useless in most case because
mass law if I use the frequency that is
far beyond the resonant frequency of this partition, then it follows mass law.
So we saw interesting picture like this.
Follows mass law over here.
Plot frequency follows 123 divide
by mass per unit area or that sort of thing.
I said last lecture this is the key idea, essential to understanding,
this actually opens the understanding about the partition.
Noise control problem.
All right, back to this problem.
Then again the transmission
will be induced by the radiation of this kind of condition.
And tau I will recall is proportionate
to fluid loading.
This is an interesting formula.
Actually I got this formula from and
for linear acoustics.
This will holds for every case.
And move on to the finite plate case but
we just studied the transmission for infinite case.
But I argued that transmission loss of
infinite case certainly predicts upper bound
Of all finite case, of plate case, therefore it is rather safe
guideline to use transmission loss based on infinite plate case.
Okay, we have a remaining sense.
Sound does not impinge to the flat
surface of this, in normal instance,
or have some angle zeta i only.
I mean, sound will
Arrive at the flat surface of this continuity in many different directions.
So how can you use this result to the case of which we have
random instance case, in other words?
This is normal instance case.
Random instance means that sound comes from everywhere.
How can we use this result to the random instance case?
Conclusion?
We just invited some traction factor, which is not big.
Because think about it this way.
If they have this instance case.
Major radiation will be induced by the component abnormal to the surface.
So that would be the major contribution.
The correction factor due to the presence of this kind of wave
in this case, very negligible.
This case, there would be some contribution.
So in the text you can find the several
rarely practically used for correction factor.
You can use it.
Or if you are very brave and if you have enough time you may have assimilation.
Or you may try to get the transmission coefficient for
every instant case.
So that's the end of our lecture.
This summarized what I have taught.
So let me see what we had before.
Well, let me summarize.
We begin with Snell's law.
And because what the Snell's law physically says,
it has to be matched on the boundary in such a way
that the fluid of particle arranged over here
has to move in the manner that the motion of the fluid
of particle in this part and that part has to be same.
In x direction, not y direction because we assume the.
So we look at these Snell's Law in terms of wave number domain and
advantage of looking at Snell's Law in wave number domain is certain exhibit.
So-called exponentially decaying wave in x direction.
And then we move to the case for
the plate for having plate
as at a discontinuity.
And we look at the mathematical formulation that determines how much
transmitted will be induced compared with the instant way.
And we found that this kb has to be same as
the wave number of freed particle in this direction.
That has to be.
As well as the wave number of transmitted wave in this direction too.
And also we assume that this wave
Zeta which is a function of y in this case.
That has such an amplitude and
the propagation look like exponential minus j omega t minus kb y.
Using that assumed solution to
the plate equation, we obtained interesting dispersion relation.
Physically the wave on a plate according to that dispersion relation,
says the long wave propagate slower than short wave.
And high frequency wave propagate faster than
Low frequency wave.
And we.
Argued that using.
What I'll draw in this picture.
Infinite plates case.
And a finite plate case that transformation curve and a finite plate
has to be larger than infinite plate.
Therefore, per unit area, transmission loss,
infinite plate case would be greater than equal to the finite plate case.
Therefore our prediction based on infinite plate
would be save those control guidelines.
To understand why we got a more transmission curve for final plate case.
Seeing what's going to happen.
As if we are fluid particle then we can understand
because if you oxalate this then the fluid over here and
over there will more effectively influence the motion of the plate.
And then I emphasize the gain.
Every transmission curve shown to here is in terms.
It can be expressed in terms of loading impedance and partition impedance.
For the case of bending wave, partition impedance has two component.
One is LM4 partition corruption that is minus j omega m.
The other one, what is associated with bending rigidity?
That's what I'm going to say today.
