We've talked a little bit to this point now about wave propagation.

We've discussed the concept of resonance and we've talked a little about room

acoustics. And, and our room acoustics are important

because they actually color, if you will, the sound that you hear.

but let's talk now as we begin to think about future opportunities for design.

Maybe in loudspeakers or, or you know, the project with the class.

We have opportunity design, to design a guitar amplifier.

But let's take about the transducer itself, which is the loudspeaker that

radiates. Before we do that though, we should first

discuss a piston radiating in a infinite baffle and this concept of infinite

baffle. It is important, and I'm going to explain

that in just a second. But basically, what we have is a piston

with some radius, A that's going to be vibrating.

so it'll oscillate in and out of this infinite baffle, which is here in blue.

and it'll go on and on in both dimensions.

The only reason that we model this as an infinite baffle is to, to separate the

sound waves that are coming off the front side of the piston.

From the sound waves that are coming off the back side of the piston.

Because these sound waves, when the piston is pushing, if you will, the air

in this direction, it's actually pulling in the opposite direction.

on this side the waves would be compressive, on the other side it's

what's known as refraction. and so as it oscillates.

And then, of course, when you have the piston pushing in this direction, you

have compression on that side of the baffle, and rarefaction on this side.

So, you know, the bottom line is is that we put the baffle here to separate these.

Because otherwise, at low frequency, you'd just have cancellation.

Because the sound wave on the back of the piston is out of phase with the sound

wave on the front of the piston. this becomes important [COUGH] in a

speaker. And what I've done here is I've just

sketched a simple speaker box with a transducer here.

But, of course, we get radiation into the listening environment.

So, you know, we may be over here enjoying the the sound that's coming off

the pist, off the speaker. But if we didn't enclose the back of the

speaker, this transducer in this box, then the sound waves inside the box would

be propagating as well. And at low frequency, they would actually

cancel. much of the sound wave.

If you remember, at the beginning of the course, I pulled the transducer out of

the box. And when I did that, low frequency

response disappeared. And that's because we had the

cancellation of the sound pressure wave from the front and back of the speaker.

So, bottom line is, is the box effectively serves, the enclosed box

effectively serves and an infinite baffle for designing speakers.

Alright, so um, [COUGH] we can derive the response of the piston vibrating.

And the radiation characteristics of that in, in the acoustic space.

again, this is another thing that's beyond the scope of the course itself,

but it's, it's still worth discussing briefly.

So what we're going to assume that the piston vibrates normal to the baffle,

basically in the z direction as we see here in the in the sketch.

So, we have our piston here vibrating normal to the z direction.

We describe some coordinates associated with the center line here.

We have a radius A of the piston we have an angle theta here at and an angle phi

here. But we can write an expression for the

pressure. As a function of the radius theta and t.

So, we are going to look at the, this area here straight out normal to the the

piston is known as on-axis. And as we move out in theta, that's known

as off-axis response. And one of the assumptions that we're

going to make is that the pressure we choose to observe is in the far field.

It's far more complicated I'll, I must say in the near field, meaning when

you're very close to the piston and you're measuring the pressure.

But if you move into the far field where the radius or the distance from the

speaker or piston is much greater than the radius of the speaker itself.

Then, the expression becomes a little more manageable.

or a little, it's more simplified, it's great, it's more greatly simplified is

what I'm trying to say. Okay, so here's our expression for the

pressure as a function of the distance from the vibrating piston itself.

the, the angular rotation of axis, and of course time.

And so, you see, you know, the typical parameters that we've had earlier.

which, you know, are density, the sound field again, the distance from the piston

here. this H of theta, which you see in the

equation is known as the directivity function..

And this is going to help us define what the response looks like off axis, the

alpha, theta. And as you can see, it's a function of

the product of the wave number and the radius of the driver.

And then, of course, a actually a Bessel function here as well.

We Mark Buckel/g, my colleague, is going to provide an overview of the Bessel

function. And, you know, you shouldn't stress over

Bessel functions. I's just the first time you were

introduced to a sine or cosine function you probably thought that was a bit

strange. I would put the Bessel function in a

similar category but we'll pause for a second now and I'll let Mark explain the

Bessel function. So we we're talking a little bit about

Bessel functions, and I thought it might be good to provide you with a reference

for that. you can certainly look them up under

Wikipedia which I've I've captured a component of the screenshot here from the

webpage. And you can see the you can see the, the,

the webpage link right here, okay? we're dealing with Bessel functions of

the first kind and in particular for the directivity pattern alpha equals one in

our case. there's a fairly complex expression here.

let's not worry about that for the moment, let's just talk a little

generally about them. I don't want you to concern yourself, in

some ways they're very similar to sine and cosine functions.

I mean, when you first heard about those, you didn't know what they were either,

possibly. For those of you who haven't heard about

Bessel functions, just think of it like a a kind of another defined function.

Like like those that you learned about when you were studying sins and cosines.

And you can see here, in particular, for integer value of n, and again remember

our, for our case n equal 1. you can use an integral representation.

And, in fact, the definition of the function is defined in terms of cosine

and sine functions here. It's an integral of that.

And I think the graphic over here shows quite clearly that the you know, the

Bessel function corresponding at n equal 1.

And you can see the the oscillatory behavior that's very reminiscent of the

sine and cosine waves. But that's also the the very kind of

directivity pattern that we saw when we were looking at the at the radiation from

from a radiating piston. So anyway, this is a little bit about

vessel functions. You certainly, if you have the

background, you can delve into it as deeply as you like.

If you don't you will pick it up at some other time in studying mathematical

functions. Okay, so now that we understand a little

better what the Bessel function is we'll talk about it's limits and how that

defines their activities shortly. The last thing to to note here, again, is

that the response of the sound pressure level decreases with respect to our

radius. so decreases within in, in, inverse

relation to the radius. The distance from the the distance from

the driver itself.