Maybe you use standing wave apparatus.

You put specimen over there.

And you excite the tube with the speaker.

The waves is coming,

some way will be reflected due to impedance over there,

call this is Pi,

and this is Pr.

Okay?

Standing wave apparatus.

And you want to measure the absorption coefficient over there.

The definition of absorption coefficient is as we note before,

energy in, and observe

the energy which is- right?

So if we measure this, that means,

if I measure- then I can measure it.

What is this? This is related with the reflection coefficient, right?

So that I can write that is one minus R magnitude square.

So, if I want to measure the absorption coefficient,

then I need to measure reflection coefficient.

Okay? And the reflection coefficient is obviously related

with impedance of the space bar.

Okay, what is the assumption of using this?

The assumption to use this as standing wave is, of course,

the cross-sectional dimension has

to be much much smaller than wavelengths of interest.

Yeah. Why? Because the cross-sectional dimension of

the standing wave apparatus is

larger than the characteristic wavelengths we are going to handle,

and there will be some higher mode in the cross-sectional area.

Something wrong to me? So, for example,

if I want to measure the absorption coefficient in the order of kilohertz,

then one kilohertz corresponding to wavelength, what?

Three centimeter.

So, usually standing wave apparatus for high frequency is very small diameter,

as you can see in your laboratory.

But for the frequency order of 100, 200,

corresponding wavelength should be very large,

more than 1.5 meter, right?

Then, cross-sectional area, what

I'm saying is for measuring this kind of I mean,

if the absorption frequency for the order of one kilohertz,

we need to have the tube whose diameter should be much much smaller than this.

Then using that small diameter of the tube,

we can also measure the absorption coefficient of this.

Okay? That is possible if and only if,

the speaker, the small sized speaker can drive effectively low frequency.

But it's not true.

So we need to increase the size of the exciter or

speaker to have sufficiently good low frequency sound.

That's why you have a tube that is very

big compared with the tube that we normally use for the case of high frequency.

Okay? So, the reason why we have two different standing wave apparatus,

is just simply not because of this argument,

but because of the characteristics of speaker.

Okay, that's important in terms of measurement technique.

And what other assumption the standing wave apparatus must follow?

Of course, this one has to be rigid.

Rigid Wall.

Second assumption, wall has to be rigid.

Yeah, of course.

If wall is not rigid,

then wall indeed have some absorption,

and that would not be good.

Of course, we can calibrate it,

but at least we have to attempt to make a very rigid wall.

That's why the standing wave apparatus is very heavy and rigid.

So, if you want to design or make your own standing wave apparatus for your experiment,

then you have to pick up the material that can be regarded to be rigid.

And as you know, there's nothing really perfectly rigid.

So. What you have to do is you have to measure how much energy is

absorbed by your assumed rigid wall before you actually measure this.

Either you have to provide your own calibration sheet that

has to evaluate the rigidness of this wall.

As well as, you need to have some data that shows how

effectively your speaker can generate the sound

with the certain frequency of interest you want to have.

That's another point of interest.

Now, how to measure R. So,

let's see the pressure we will measure.

Total pressure will be Pr plus Pi,

where Pi is the wave

that is propagating in this direction,

so I will use the jkx,

but we are using exponential minus j omega t,

so that would be minus jkx.

In our tax table, I'm using this.

So, for left going wave would be minus jkx plus omega t,

therefore I'm using this because this is the wave propagating in this direction.

And let's say P0 is the amplitude in complex of the instant wave.

And then, Pr will be P0

exponential of jkx in this case because Pr is propagating in this direction,

therefore it should be exponential minus jkx minus omega t,

plus jkx minus omega t. Then I use the same minus omega t over there.

So, I'm using plus jkx,

and this will be reflect or defined by the reflection coefficient R. And obviously,

R has a real part and imaginary part.

And our objective is to figure out

how this one is related with the absorption coefficient.

And this is it.

And then now, our objective is how to measure

R. Inside of there,

what we're measuring for example,

I'll put microphone over here,

what we measure is P and X.

But I want to measure R from P. Okay, let's rewrite P,

then P is equal to one is the due to distance,

so that is exponential minus jkx,

and the other one is RP0 plus jkx.

Therefore, I may write and this is

P0 exponential minus jkx plus R exponential plus jkx.

From this mathematical expression,

what you can see,

the inside of that standing wave,

there is some sound fluctuating.

Maximum and minimum.

So, if I measure P at certain point say,

I measure at x equals zero,

then what I get is P0 one plus

I measure P,

then what I get is one plus R,

and I do not know P. So,

maybe I select another point.

Then I have two equations,

and I have two unknowns,

P0 and R, and I can get it.

But because this is experimental approach,

we have to worry about signal to noise ratio.

What you measure inside,

there would be some noise.

So, the very wise way to maximize the signal to noise ratio would be,

I measure P max,

and I measure P minimum.

Measuring these two values,

and then calculate R. That's one way.

And how we do it, inside of two,

due to the reflection,

there would be some peak,

minimum, and then peak and minimum,

and then peak, things like that.

And you pick up this point and that point and measure alpha.

And then you pick up another point,

and the measure alpha,

and you pick up another point and measure alpha, then average it.

Because average will reduce the effect of noise, because if you

have a single note like that,

if you average it,

you will get to this mean value,

and if the signal has some noise with the C component,

then if you measure it,

you will have a true value.

And for this case,

what you get is specimen.

You have some maximum minimum, and maximum minimum,

and maximum minimum, and this is induced by some noise.

Therefore, you got estimate alpha one,

and using that value,

estimate alpha two, estimate alpha three.

And what you do is,

alpha one plus alpha two,

plus alpha three divide by three,

and this you average absorptions coefficient.

And you do similar things in every frequency of interest.

So that means you need a long tube to have many data points.

That's why in your level,

I have to use the two in very long size like this size.

So, measuring absorption coefficient is not true,

but you'd have to think about many different things.

Lecture 4-2 Part 2. Sabine's Theorem 2

Introduction to Acoustics (Part 2)

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