Properties of periodic waves

in the last video we talked about the idea that if I start with some type of a string there and if I were to take the left end of the string I could just as equally have done the right but if I take the left end of the string and jerk it up then all the way down and then back to its resting position it'll generate this disturbance in the string and the disturbance might initially look like this after I've done that jerking up and down once and that disturbance is going to propagate down the string it's going to move down the string like that let me color this in in black so this is right after I do that first cycle that first jerking up and down the string might look something like that and if we wait a little while the string might look something like this assuming that that's I only did that once the string might look something like this where that pulse has actually propagated down the string that pulse has propagated down the string and we in the last video we said hey this disturbance that's propagating down the string or propagating down this medium although it doesn't necessarily have to have a medium we call this we call this a wave and in particular this wave right here this is a pulse this is a pulse wave because we only have essentially one perturbation of the string now if I kept doing that if I kept going up and down and up and down essentially if I periodically did it at regular intervals then my string would look something like this it would look something like this it would look something doing my best to draw it neatly it might look something like this where once again the perturbations are going to where the disturbances are going to move to the right they're going to move to the right with some velocity and what I want to do in this video is really focus on this type of wave this type of wave right here which you can imagine since I'm periodically moving this left side up and down up and down and creating these periodic movements in the wave we call this a periodic wave this is a periodic periodic wave the motion is repeated over and over again so what I want to talk about is some of the properties of a periodic wave now the first thing you might say is well hey you know how far are you jerking it up and down how far are these movements from rest so if this is kind of the resting position right there how far these movements up and above the resting position and below the resting position and we call that the amplitude of the wave so that distance right there I'll do it in magenta that distance right there is the amplitude amplitude amplitude sometimes Mariners will have an idea of wave height wave height they normally refer to from the bottom from the trough of a wave to its peak amplitude we're talking about from the resting position to its peak so let me label peak well I think you know what peak means peak is the highest point on the wave that's the peak and that's the trough if you're in a fishing boat and you wanted to see how big a wave is you'd probably care about the wave height not so much the L you you know if you're both sitting down here you have to care about this whole distance but anyway we won't talk too much about that so that's kind of the first interesting idea behind a wave is G and you know not all waves are being generated by Sal jiggling a string on the left-hand side but I think you get the idea that these waves can represent many different these this graph can represent many different types of wave forms and this essentially displacement from the resting position or from the zero position that is your amplitude now the next question you might ask is okay I know how far you're jiggling this wrist this string up and down but how quickly are you doing it so how long does it take for you to go all the way up all the way down and back again so how long for each cycle we could say you know a cycle is me going up down back again how long for each cycle or you might say how long for each period right we're saying this is periodic each period is each repetition of the wave so this idea of how long for each cycle we call that the period we call that the period and this is going to be a unit of time maybe I'm doing it every two seconds it takes me two seconds to go up down back again up down back again that's going to be two seconds now a very related term is is how many cycles am i doing per second how many cycles cycles per second so notice this is how long or you could say how many seconds for each cycle we could even write that assuming we're dealing with second how many seconds for each cycle so for example this you know a period it might look like something like five seconds per cycle or maybe it's two seconds per cycle but then what if we're asked how many cycles per second so we're asking the opposite question and not how long how many seconds does it take for me to go up down and back again we're saying in each second how many times am I going up down back again so how many cycles per second that's the inverse of period so period the notation is normally a big capital T for period this is frequency this is frequency frequency frequency it's normally denoted by an F and this you're going to say cycles per second so if you're going five seconds per cycle that means you're doing one fifth of a cycle per or 1/5 of a 1/5 of a cycle per second all I did is invert this that right there and that makes sense because the period and the frequency are inversions of each other this is how many psych seconds per cycle how long does one up down back again take and this is how many up down back again are there in a second so they're inverses of each other so we could say that frequency is equal to 1 over the period or you could say that period is equal to 1 over the frequency so if I told you if I told you that I'm I'm vibrating the left end of this rope at let's say it's ten cycles ten cycles per second and by the way the unit of cycles per second this is a Hertz so that I could have also written this down as ten Hertz which you've probably heard before ten Hertz just mean ten cycles per second if it's if my frequency is ten cycles per second my period is going to be one over that so one over ten it's going to be one over ten seconds per cycle which makes sense if I can in ten times I can go up and down a hole up down back again if I can do that ten times in a second it's going to take me one tenth of a second to do it each time now another question we might ask ourselves is how quickly is this wave moving in this case to the right since I'm jiggling the left end of the string how quickly is it moving to the right so the velocity so to do that we need to figure out well how far did the wave go after one cycle or after one period so you know after I jiggle this once how far did the wave go right what is this distance from this resting point to this resting point there and we call that a wavelength that is a wave length and there's a lot of different ways you can define a wave length you could view a wave length as how far the initial pulse went after completing exactly one cycle or you could view it as the distance from one peak to another peak that is also going to be the wavelength or you could view it as a distance from one trough to the other trough that's also the wavelength or in general you can view the the wavelength as one exactly equal point on the wave so exactly you know from that distance to that distance that is also that is also one wavelength where you're completing between that point in that point you're completing one entire cycle to get exactly back to that same point and when I say exactly back to that same point this point doesn't count because this point although we're in the same position we're now going down we go to the point where we're in the same position and notice over here we're going up we want to be going up again so this isn't this distance is not one wavelength to go one wavelength we have to go back to the same position and we're moving in the same direction so this is also one wavelength so if we know how far we've traveled after one period so let me write it this way so wavelength wavelength is equal to how far how far the wave has traveled after one period after one actually or after one period or you could say after one cycle or one cycle because remember a period is how long does it take to complete one cycle one complete up down and back again motion so if we know how far we've traveled and we know how long it took us it took us one period how can we figure out the velocity well the velocity velocity is equal to is equal to distance divided by time is equal to distance I'll write it out is equal to distance divided by time well for a wave your velocity and I could write it as a vector but I think you get the general idea your velocity what's the distance you travel in a period well the distance you travel in a period is your wavelength is your wavelength right after one after one up down back again the wave pulse would have traveled exactly that far that would be my wavelength so I've traveled a distance of a wavelength and how long did it take me to travel that distance well it took me a period to travel that distance so it's wavelength divided by period divided by period now I just said that one over the period one over the period is the same thing as the frequency so this could be I could rewrite this as wavelength and so actually I should be clear here the notation for wavelength tends to be the Greek letter lambda so we could say velocity velocity is equal to wavelength over period which is the same thing as wavelength times one over my period and we just said that one over the period this is the same thing as your frequency so weight velocity is equal to wavelength times your frequency and if you know this you can pretty much solve all of the basic problems that you might encounter in waves so for example if someone tells you that I have a velocity a velocity of I don't know 100 meters per second to the right so in that direction velocity you have to give a direction and they were that they were to tell you that my frequency my frequency is equal to I don't know let's say there my frequency is 20 cycles per second which is the same thing as 20 Hertz so literally if you were if you were only to see this if you had a little window where you're only able to observe this part of your wave you only observe that part of my string if we're talking about 20 Hertz then in one second you would see this go up and down 20 times 1 2 3 4 5 6 7 8 9 10 11 everything is then 19 20 that would happen in 1 that is exactly 1 second you would see this go up and down 20 times that's what we mean by the frequency being 20 Hertz or 20 cycles per second so they gave you the velocity they gave you the frequency what's the wave length here so the wave length in this situation you would say the velocity let me write it over here your velocity is equal to your wave length times your frequency times your frequency divide both sides by 20 and actually let me make sure I get the meter the unit's right so this is meters per second is equal to lambda times 20 cycles cycles per second so if you divide both sides by 20 cycles per second you get 100 meters per second times 1 over 20 seconds per cycle and then you this becomes a 5 this becomes a 1 so you get 5 and then the seconds cancel out and seconds so you get five meters per cycle so this is equal to five meters per cycle which would be your wavelength in this situation so it's five meters you could say five meters per cycle but wavelength implies that you're talking about the distance per cycle so in this situation if this if this is moving to the right at 100 meters per second and this frequency I see this moving up and down 20 times in a second then this distance right here must be five meters likewise we can figure out the period very easily the period here is just going to be one over the frequency it's going to be one over 20 seconds seconds per cycle so using this you know these formulas and I don't want you to memorize the formula I want to be able to kind of it should be intuitive for you and hopefully this video made it a little bit intuitive but using this you can really answer almost any question if you're given two or two of these variables and you need to solve for the third anyway hopefully you found that helpful