Wave Functions & Equations of Waves

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

Hey, guys, let's do a quick example. Draw the displacement versus position and the displacement versus time graphs of a trans verse wave given by the following representation. And then they give us the mathematical representation of that way. Okay, I'm gonna draw the wave over a single period because there's no way to draw all of the waves. These waves go on from negative infinity to positive infinity. They go on forever. So I'm just gonna draw a single period. You might be asked to draw two periods of the way over three periods or four periods or whatever. But if you're told just to draw the wave, draw a full cycle doesn't really matter how much you draw. So here my graphs. Here's displacement versus time. Here's displacement versus position. In order to graft them, we need to know three things We're gonna need to know the amplitude. Okay, We're gonna need to know the period, and we're gonna need to know the wavelength. Okay. Amplitude is easy, right? 1.5 centimeters. Amplitude done. So on both of these graphs, I'm gonna write 15 centimeters and negative 1.5 centimeters. And these were gonna mark the boundaries that the leaves air going to oscillate in between the waves. You're gonna stay between the amplitude. Now, remember, this number right here represents the wave number, and this number right here represents the angular frequency. Don't forget that. So our wave number is in verse. Centimeters and that's two pi over Lambda. So Lambda is two pi over which is about 30 centimeters. Okay, Now the angular frequency, as we can see, is two pi over 0.1 seconds. And the angular frequency related to the period is two pi. Sorry, little technical difficulty. Two pi over the period. So relating these two equations together, we can see that the period is simply 20. seconds. Okay, so now we have enough information to draw. Ah, full period or a full cycle of each of these waves. We know that during the cycle, the wave is going to take 0. seconds. So if I draw 0.1 seconds right here on my displacement versus time graph, I can just draw my sine wave. Right? This is a sine wave. If this was CO sign we would start at a amplitude, drop down to the negative amplitude and go back up to the amplitude. Okay. For displacement versus position, we know that a cycle takes three centimeters, so I'm gonna mark three centimeters and I'm gonna draw the same Yeah, sine wave. Okay. And this is exactly the position. Sorry. The displacement versus time and the displacement versus position graphs for this function. All right, guys, Thanks for watching.

Hey, guys, In this video, we're going to delve a little deeper into the mathematical representation of waves and cover a related concept called phase angle. Okay, let's get to it now. We said that a wave that begins with no displacement is a sine wave, right? And a wave that begins with a maximum displacement. A displacement at the amplitude is a cosine wave. But what if we have a wave that begins with a displacement in between the two? Then what happens? Well, in this case, the most complete description of a wave. The best mathematical description of a wave is to combine all of those possibilities in tow one. And we typically right, That is a sign K X minus omega T plus fi, where fi is what we call the phase angle. Okay, the phase angle tells us what that initial displacement is. Okay, the phasing was determined by the initial displacement off the wave. Ah, wave that begins with no displacement has a phase angle of zero degrees, which is a pure sine wave. A wave that begins with a maximum displacement has a phase angle, power to which is a pure cosine wave. Um arbitrary wave One has displacement between zero and its maximum. It's gonna have a phase angle between zero and pi over two. Okay, It's gonna be a mixture of sine and cosine waves. Let's do an example away with a period of seconds and a velocity of 25 m per second has an amplitude of 12 centimeters. At T equals zero and X equals zero. The wave has a displacement of eight centimeters. What is the mathematical representation of this wave? Okay, so we're saying, why is a sign K X minus omega T plus five? Okay, since X and tear both zero, we can ignore them. And all that PFI depends upon is the initial displacement. So this is going to be 12 centimeters sign of five, and that equals eight centimeters. That's the initial displacement. So sign of Phi equals eight centimeters over 12 centimeters. And FYI, which is the inverse sign of 8/12 is simply Okay, very, very quick. Very easy. All right, guys, this wraps up our discussion on the phase angle and the proper representation off a wave as a mathematical function. Thanks for watching

Hey, guys, in this video, we wanna talk specifically about the velocity of a wave. Okay, let's get to it. Remember, guys, that there are two types of waves, right? We have trans verse, and we have longitudinal waves. Both of them have a propagation velocity. Both waves move right there, both of them propagate. But trans verse waves have a second type of velocity, which we call a trans verse velocity, which is how quickly the medium is moving upwards. For instance, a wave on a string Ah, wave on a string is not only moving horizontally at some propagation speed, but the individual parts of the string are also moving up and down with some velocity. That's the trans verse velocity. The propagation velocity of a wave depends upon two things. It depends upon the type of wave, and it depends upon the medium that the wave is in. This is an absolutely fundamental property of all waves. Okay, the type of wave will tell you the equation to find the speed, the medium will tell you some number that goes into the equation. All right. And we're gonna get Teoh a type of wave on a sorry type of transverse wave called waves on a string where we will see that the type of wave idea wave on a string will determine the equation and the medium i e. The tension on the string and the mass pregnant length of the string will determine how fast it goes. Okay, so it's both. It's the type of wave, and it's the characteristics of the medium that it's in. The only way that does not propagate in a medium is light. All other waves propagate in the medium like can propagate in a vacuum. Okay, we can rewrite as I mentioned it, a problem before we can rewrite the equation for a wave like so. This is using the fact that K is two pi over Lambda and Omega. I'll be right that different color and Omega is two pi over tea where t I can relate Thio of speed using our regular old speed equation. And this becomes two pi over Lambda Times V so I can pull this two pi over Lambda out of the equation and I'm left with a V here. This form tells us the speed of the wave instantly and the direction that it's going in when we have X minus VT as our input, the wave is propagating in the positive direction when we have X plus v t as our input. Where V is a positive number in both of these cases, the waves propagating in the negative direction. Okay, a wave is represented by the equation given here. What is the propagation velocity? Is it positive or is it negative? Okay, A really quick way to find, given the mathematical representation of a wave, the speed of the wave is that the propagation speed is always going to be omega over K. Okay, this is a really quick sorry equation that you can show is true very easily using these substitution ins that I said omega is two pi V over Lambda over two pi over Lambda. Those two pies over Lambda are left. Sorry, they canceled and all that is left is V okay? You guys don't have to write that down. By the way, I'm just showing you that it's true. All right, Now the equation is written where we have a mega right here and we have k right here. So mega is 1.7 inverse seconds. Que is point to inverse centimeters, which is? Uh huh. 85 centimeters per second. Okay, Now the question is, is it positive or is it negative? Well, this is a negative sign. So it is positive. Okay, Now, on a trans verse wave, we have a trans verse component of the velocity because the wave is going up and going down and going up and going down and going up and going down. Obviously, the velocity is changing that trans verse velocity, right? Initially, it's going up. So it's a positive velocity. Then it's coming down, so it has a negative velocity. Since that's changing, there must be a trans verse acceleration. Okay, Okay, So it's another thing to worry about in the transverse direction, Given our general equation four. Ah wave. We have general equations for the trans verse velocity and for the trans verse acceleration in general, the trans verse velocity can be written as the amplitude times omega cosine of K X minus omega T plus fi. Whatever the phasing will happens to be, in general, the acceleration can be written as negative. A omega squared sign K X minus omega T plus five. None of the numbers change a omega K five. They're all the same in these equations as they would be in the irregular equation for the wave. Right where we have. Why is a sign K X minus omega T plus five. It's all the same variables. All right Now, co sign can get as biggest positive. One in a small is negative one. Obviously, the maximum trans verse speed is just a Omega likewise sign can get his biggest positive one in the smalls Negative one. So the largest transverse acceleration is just a omega squared Just those coefficients of the trick functions. And lastly, we want to do one more example. Ah, Longitudinal wave has a wavelength of 12 centimeters in a frequency of hertz. What is the propagation speed of this wave and what is the maximum trans verse velocity of this wave? Okay, well, we'll use our regular old regular old wave speed equation. This is the propagation speed, right? It goes forward. Some distance, Lambda, in a period times the frequency of the wave. Okay, so as a wavelength of 12 centimeters and the frequency of 100 hertz which is 1200 centimeters per second or 12 m per second. Okay. And what about the maximum trans verse velocity? Well, there is no transverse velocity. This is a longitudinal wave. Longitudinal waves. Yeah. Okay. Thank you. Longitudinal waves have no trans verse velocity. The transverse velocity is always zero for longitudinal waves. Right? There is no trans verse oscillation. All the oscillation occurs down the length sorry. Down the propagation distance of the wave, there's no transverse component. So there is no transverse velocity for longitudinal waves. But they're absolutely is still a propagation velocity for the wave. And it follows the exact same equation that you would use for a trans verse wave. Alright, guys, that wraps up our discussion on the velocity of waves. Thanks for watching.