The Mathematical Description Of A Wave

by Patrick Ford
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Hey, guys, in this video, we're gonna talk about the actual specific mathematical description of a wave instead of just talking about attributes to a wave like the period or the wavelength of the speed we're gonna describe away fully with a single function. Okay, let's get to it. Since waves air oscillate story, we have to use oscillating functions to describe them. These are our trig functions, okay? Our signs and our co signs. All right, We're not gonna worry about the other trig functions because they aren't truly oscillating functions. Okay, the displacement that the way begins with or the initial displacement is going to determine the type of trick function, whether it's gonna be a sine or cosine. Okay, let's take this first wave on the left. It begins at the origin, and then it increases decreases, etcetera. This we know, is a sine wave. Okay, so this wave will be described by a sine function. Now the wave on the right, the wave right above me begins initially at the maximum displacement at the amplitude. Then it decreases and increases and decreases etcetera. This is a cosine wave. Okay? And so the mathematical function, describing that wave is going to be a cosign now. Both of these graphs show a displacement versus time, right? These air both oscillations in time. Now simple harmonic motion is described by oscillations in time, and you would have very similar functions, signs or co signs to describe them. But waves propagate in space, so we cannot Onley describe their oscillations in time. We have to also describe their oscillations in space. Now, if the wave happens to be a sine wave in time, it's also a sine wave in position. And the same applies for Cozzens. Okay, now waves like a set of more properly described in terms of oscillations both space and time. So let's do that now for a sine wave, we would say it has some amplitude times Sign of K X minus omega T. Now I'll tell you what K and Omega are in a second, but this has both space dependence or position dependence and time dependence exactly like we want or for CoSine wave. We've had a co sign K X minus omega T. Okay, so the question is, what's K on? What's Omega Que is something new that you guys haven't seen before? called the wave number. Okay, Where to? Pie divided by the wavelength. Alright, Omega is something you guys have seen many times before. It's simply the angular frequency or two pi times the linear frequency. Okay, let's do a quick example. A wave is represented by the following function. Mhm. What is the amplitude? The period, the wavelength and the speed of the wave. Okay, so we want to gain all that information from the single equation that describes this waves oscillations in both space and time. Okay, so first, this equation isn't quite of the form that we have seen before. We want to take this coefficient right here on. We wanna multiply it in words because we want to have our equations of the form. Why equals a cosine K x minus Omega T. Okay, so we need to multiply this number inside. So we confined readily what K is and what Omega is. Okay, so this is going to be wide equals 0.5 cosine off. Well, exes coefficient right now is one. So it just gets the two pi over 10 which is 0.6 centimeters in verse. The coast. Sorry, the coefficient of tea is seven. So we need to do seven times two pi over which becomes 8 in verse seconds. Now, really quickly Notice that the units of seven or meters and the units of our number all the way on the outside are centimeters. Those need to be the same unit to cancel once you create in the same unit, either both of them centimeters or both of them meters. Then you will find this 439.8. If you don't convert, you're gonna get 4.398 which is the wrong number. Okay, so just make sure that you convert notice right off the bat that we confined the amplitude we confined at the wave number and we confined the angular frequency just by looking at the equation. So the amplitude done the way the nothing else. We need to use these to find the wavelength, the period and speed. Okay, so the wave number is related to the wavelength. All I have to do is multiply pie up and divide the weight of number over. So this is the wavelength is two pi divided by the wave number, which is two pi divided by 20.6 to 8, which is going to be 10 centimeters. Okay, so that's another one that we're done. The period is related to the angular frequency. We could say that the angular frequency is two pi over the period. So if we multiply the period up and the angular frequency over in the period is two pi over the angular frequency, which is two pi over 4 39 8, which is going to be 700. Sorry. Reading the wrong part of my notes here. 0.14 three seconds. Okay, that is the third thing that we need to find. Finally, we need to find the speed, but we know the wavelength, and we know the period, So the speed is easy to find. The speed is simply the wave length divided by the period, which is going to be 10 centimeters divided by 100.143 seconds, which is gonna be 700 centimeters per second or 7 m per second. Okay. And that is all four things that we were asked to find. Okay. Really quickly, guys noticed this number seven right here. 7 m per second. And this number Sorry. 10 centimeters over there to the left. Those numbers appear here and here. Why do they appear there and there? That's because the equation off the form that it's written I'm gonna minimize myself and put a little note right here. The equation off the form that it's written is why equals a cosine of two pi over Lambda X minus V t. This is another very common way off writing the mathematical equation for a wave. Okay. And if you notice you already have Lambda and the Speed written right, Landis 10. The speed is seven. Okay, but since we didn't cover this explicitly, we didn't cover this explicitly. I didn't want to start from that point. Alright, guys, that wraps up our discussion on the mathematical description of a wave. Thanks for watching