This week, we're going to talk a little bit about how we convert electrical signals into sound. It's done with loudspeakers, obviously. And sometimes, the term loudspeaker is used interchangeably with the transducer or what we call the driver itself. We're going to talk a little bit about that. The simple model for that in acoustics is a piston radiator, which is a perfectly flat, round plate that we think about oscillating in a boundary that's infinite. And the reason we think about that is it separates the front radiation of that plate from the back. that boundary actually is realized physically in the world through a cabinet. which we'll talk about cabinet design later. But by the end of this lecture, you should have some understanding about how sound radiates off of a speaker. And in particular, why we choose transducers with different diameters for different frequency ranges. 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.