All right. few other parameters. Of course, c is the speed of sound in air. j is a complex number, square root of minus 1. we have our radius of our piston and our wave number. And now, what I want to do is talk specifically about this component of the relationship between the wave number and the radius of the piston itself. if the product of the wave number and radius of the piston are much less than 1, then pressure is nearly independent of theta, okay? And why is that? It's because of a nature the jet vessel function if ka is much smaller than 1. Then, the vessel function of this particular argument here, ka sin theta, divided by ka sin theta, approaches 1 half, alright? And so h of theta, which is 2 times that, will approach 1. what that leads to is basically what is called omni-directional radiation. However, if the product of the wave number and the radius of the piston radiator is much greater than 1. And the source could, is going to become very directional due to what we call pressure nodes. there are many solutions to the directivity function for which the vessel function is equal to 0. And we can let those solutions, those, those roots of that equation, be represented by little j1 representing our Bessel function of the first kind and m being the mode. again, we will go back to the concept of modes, because we have a number of modes where that's zero. And so, these arguments, ka sin theta m, represent all the solutions where the Bessel function is equal to 0. And we can find an infinite number of solutions sorry. and that, that infinite number of solutions represents our modes just as we had done earlier when we were looking at modes of runes and otherwise. Before we discuss this in greater detail, I would like to go back to an animation off of Professor Russell's page. Again another great animation for a baffled piston, and here you know, you'll see the equation that represents the response of the baffled piston. We've already discussed this a bit he's given a representation where ka the product of the wave number and the radius of the driver are less than 1. And you see what's called the directivity pattern here this is our, this measure, or this directivity function that I talked about. And we get this very uniform radiation, as you see. We get almost this perfect hemisphere, meaning that the sound pressure level radiated normal to the vibrating piston. Here's the vibrating piston, is basically the same as it is at any angle off axis under these conditions. but as you go to medium frequency. Where the product of the wave number and the radius are greater than 1. Then, you basically start to get into cases where, you know, the radiation on axis is maximal here, obviously. But as you move off axis at various different angles, the sound pressure level is decreasing. And in fact, you know, in the plane here, there's no response at all. again, this is a result of the directivity function and effectively you, you're solving for these nodes, these pressure nodes that exist, okay? If we if we move to what's called a higher frequencies where the wet product. The wave number and the radius of the driver are much greater than 1. And you're going to have multiple pressure nodes, so you'll still have, you know, your maximum radiation on axis here when theta equals 0. that's always going to yield the same solution. but as ka becomes much greater than 1, you're going to have many other pressure nodes. And it becomes very directional. So this is the directivity of the radiation from he driver itself. So, you can imagine if you were listening to music through a vibrating pi, if I listen to music with a vibrating piston or your loud speaker if you were on axis. If you're sitting right in front of the speaker right in line with the speaker, you're going to have you know, good sound pressure generation. But you could depending upon this relationship here, you could get into situations where depending on where you were you were relative to the speaker. If you were over off at this angle, you'd hear no sound pressure level at all at, at a given frequency. so this gives you some good pictures of what the directivity pattern looks like. And then here he's demonstrated some measured data for a real loud speaker, okay? So, he's got a speaker in an enclosure, and he's showing the response from 250 Hertz Up to 10 khz. So, there's three response characteristic or directivity patterns measured. Starting at 250 Hertz the lowest frequency you can see a very a spherical radiation pattern around the speaker. As you move to 2500 Hertz, you can see that the speaker radiates on axis more, but it's still fairly uniform even off axis in this area. by the time you get to 10 kilohertz, it becomes far more directional in terms of the radiation. So this ends up being you know, an important way to characterize the, the response of the speaker. And it becomes an important part of how we design speakers. So, you know, the first question you might ask are, are, why are the transducers different sizes? You know, the in the loudspeaker cabinet, we have multiple speakers or transducers here. this one's typically called a tweeter and sometimes this is a, a woofer. and so the question might be is why, why are these transducers different sizes? Well, I think you probably know the answer to that. And here we sketched what is a a two way speaker. the small transducer is really for generating high frequency sound radiation, and the larger driver is is there for for low frequency sound radiation. And, you know, if we go back to this discussion about the product of the wave number. And the radius of the driver, if it's much less than one we have omni-directional radiation of sound. So, the issue you have with the woofer and the reason you can't really use that to generate your high frequency sound, is that if you want to have a good radiation pattern. you know, something that's much broader, then you're going to need smaller-diameter or smaller or drivers with smaller radius to achieve that. And so, I've sketched that here. I mean, this is what we're really looking to accomplish. We're looking to find a radiation pattern such that you know, the sound pressure level response off-axis looks very much like it does on axis. And, one driver, a very large driver will start to become very directional, you know, will get characteristic patterns like that maybe as we saw in the sketch earlier. as you move to higher frequency, and you can get far more complex in terms of the nature of response. So, bottom line is, is that the transducers of different sizes allows us to get a much more uniform variation pattern from the from the speakers.