Velocity Of A Wave

by Patrick Ford
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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.