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Video transcript

if you've got a medium and you disturb it you can create a wave and if you create a wave in a medium that has no boundaries in other words a medium that's so big this wave basically never meets the boundary then there's nothing really stopping you from making a wave of any wavelength or frequency whatsoever in other words there's not really any naturally preferred wavelengths they're all pretty much as good as any other wavelength however if you confine this wave in to a medium that has boundaries this wave is going to reflect when it meets the boundary and that means it's going to overlap with itself and when this happens you can create something that are called standing waves we'll talk about what these mean in a minute but the reason we care about them is because when standing waves happen they select preferred wavelengths and frequencies only particular wavelengths and frequencies are going to set up these standing waves and what ends up happening is that these often become dominant and that's why these standing waves are important to study so let's study some standing waves let's take a particular example let's say you've got a string of not that many strings one string here and you nail this string down at both ends so you're going to prevent any motion from happening at the end of this string the string can wiggle in the middle but it can't wiggle at the endpoints and this isn't that crazy a guitar string is basically a string fixed at both ends piano strings or strings fixed at both ends so the physics behind standing waves determines the types of notes you're going to get on all of these instruments and by the way this point over here we're basically making sure that it has no motion so by nailing it down what I really mean is that there's gonna be no motion at this endpoint and no motion at this endpoint and instead of calling those no motion points physicists came up with a name for that they call these nodes so node is really just a fancy word for not moving at that point so for this string there's going to be nodes at each end and we'll see when you set up a standing wave it's possible that there's nodes in the middle as well but they don't have to be for this string though we're making sure that there have to be nodes at each end so why does a standing wave happen and how does it happen let's say you give the end of the string here a little pluck and you cause a disturbance that disturbance is going to move down the line because that's what waves do it's going to over here once it meets a boundary it's going to reflect back to the left now it turns out when a string hits a boundary where it's fixed when it hits a node in other words it gets flipped over so you might have tried this before with the hose if you send a pulse down the line and you try to see how it reflects it gets reflected upside down that doesn't matter too much for our purposes but every time it's going to reflect it flips its direction and it keeps bouncing now let's say instead of sending in a single pulse we send in a whole bunch of pulses right we send in like a simple harmonic wave now when this thing reflects it's going to reflect back on top of itself because this leading edge will get reflected upside down this way and it's going to meet all the rest of the wave behind it and overlap with it creating some total wave that would be composed of the wave traveling to the right plus the wave traveling to the left and you can use superposition and interference to figure out what that is which for most wavelengths is just gonna be a mess so if you just send in whatever wavelength you want and let it reflect back in on itself the total wave you get might not really be anything special it might just be sort of a mess in here nothing really all that interesting however there will be particular wavelengths that set up a standing wave I'll show you what that looks like in a minute how do you find these special wavelengths you simply ask yourself what wavelengths could I draw on this string so that there was a node at each end what wavelength would fit inside of this region and have a node at both ends so instead of trying to add up a complicated superposition of waves we can figure out the special wavelengths simply by drawing them and seeing which ones fit so let's try it out let's get rid of all this all right what wavelengths would fit in here well we know what a simple harmonic wave looks like it looks something like this so the question we need to ask is if we start at the zero point because I want to make sure I have a node at the left end what might the shape of this graph look like so that I reach a node at the other end as well well the first possibility look at it I started a node when do I get to a node again I get to a node when it takes this shape right here there's another node so the first possibility which is going to be the longest largest possibility would be a wave that just kind of looked like this looks kind of like a jump rope this would be the first possible standing wave you could set up on this string and that means it's special it's called the fundamental wavelength this is the Big Daddy this guy dominates all the other wavelengths we're going to meet yeah there's other standing waves you can set up on here this one's the big alpha dog and if you let this string vibrate however at once it's gonna pick the fundamental wavelength and so what would we see happen the strings not just going to be suspended in air like this it's going to be moving around but these are called standing waves because this peak no longer looks like it's moving right or left this peak is just going to move up and down so a lot of times when we draw these standing waves we draw a dashed line underneath here that mirrors the bold line because all this Peaks going to do is go from the top to the bottom then back to the top is just going to oscillate it's going to look like a jump rope but it's not revolving it's just moving up then down then up then down it takes this shape then it would be flat then it might look something like this and then it comes down to here and then it goes back up and it keeps going up and down we call it standing it's more like dancing it's kind of like a dancing way but we call them standing waves because these Peaks don't move right or left so that's the fundamental wavelength what would the next possibility look like let's see we got to go from a node we know we have to go from node all the way to another node that was the first one so let's just keep going and go to the next one that would be the next possible standing wave because it'd have to fit within here what would that look like it would come up it would go down and then it would come back up so that it meets this node on the other end that would be the next wavelength sometimes this is called the second harmonic second because it's the second possibility harmonic because these are resonances and this term is used a lot when you talk about resonances with musical instruments what would the third harmonic look like what we got to start at a node we go to this one that was the fundamental this is the second harmonic so that's going to be the third harmonic so this one's going to come up it's going to go down it's going to go back up and then it's going to come back down and that would be the third harmonic and you can see you can keep going here you can create infinitely many of these but let's analyze what's going on up here what's actually happening in these standing waves note that there's going to be points like right here on this third harmonic and if I draw it's mirror so that I can get this so this is what it would look like maybe a little after actually exactly one half of a period after this bold line so you wait this peak move down here this valley moves up to this peak this peak moves down to here they're oscillating back and forth but note this point right here just stays put that's not even going to move that's a note and so is this point right here these points are happening because when those waves line back up remember the wave travels to the right bounces back to the left and at this point right here and this point right here you're getting destructive interference between those two waves similarly at these points where you're getting the maximum displacement the two waves are lining up in such a way that they're interfering constructively so the nodes are the destructive points where the wave cancels that make sense because there's nothing happening there there's no motion and these maximum displacement points are the constructive points we should give those a name what do you think we call those if you guessed anti node then you're right these are called anti nodes because that's where there's the most motion now you often have to figure out mathematically in terms of the length of the string what are the actual wave lengths you can get so drawing the picture allows you to find those but how do you actually get them mathematically well I've kind of created a horrible mess here so let me clean this up sorry about that get rid of all that let me just add some strings in here and so this doesn't get too abstract let's just say the length of this string it's pretty big let's just say it's 10 meters a really long string you secure to both ends so the first standing wave looked like this jump rope mode looked like that now if the string has a length of 10 meters what would be the wave length of this wave you might say 10 meters but not this is not one whole wavelength remember if we looked at this wave pattern we had over here that we were using this is one entire wave length you had to go all the way to here to get through a whole wave length this was only half of a wave length so this jump rope is only half of a wave length what would a whole wave length look like this would extend all the way out here all the way back up I can't really get there so I don't go off-screen that would be a whole wave length so this would be 10 meters and then other ten meters that means the wavelength of this wave even though a whole wavelength isn't fitting in here if there was a whole wavelength on the string extended this wavelength would be twenty meters what was the next wave remember it looked like this it had one node in the middle whereas this first fundamental wavelength had no nodes in the middle and again if this string is ten meters what's this wavelength equal well that's easy this is one whole wavelength so that would just be ten meters so the wavelength here would be ten meters because one whole wavelength fit exactly within the strings length of 10 meters and the third harmonic looks something like this it has two nodes in the middle note you keep picking up another node in the middle fundamental has no nodes in the middle second harmonic has one node in the middle third harmonic will have two the fourth will have three and so on so what is this wavelength this one's a lot harder for people to figure out so let's look at this one wavelength is all the way to here so this is a wavelength but our string is this long so what fraction of this length is this wavelength we'll look at this wavelength is two-thirds of the entire length of the string that means we could just say that this wavelength is two-thirds of ten meters which is we could write it as 20 meters over three and we'll keep going here I'll draw the rest this is the fourth harmonic how big is this wavelength well this wavelength covers half of the string so this wavelength is going to be half the length of the string and that's going to be half of 10 which is five meters and we can keep going I can draw the fifth harmonic down here and it would look like this and you can ask yourself how big is this wavelength well this wavelength let's see one two three four five I've got five of these humps in here so this wavelength is going to be two fifths of this entire length so I'm just going to write lambda is two times ten would be twenty meters so two fifths of ten would be twenty meters over five oh we could simplify is four meters but what if they ask you for like the forty third harmonic if they're like hey what's the wavelength of the forty third harmonic I don't want to sit down and draw like forty three of these things and try to figure out what fraction it is and you don't have to there's a pattern here so let me show you the pattern this is going to look kind of weird but I'm going to write this first fundamental wavelength as two times ten meters over one and then I'm going to write this second harmonic as two times ten meters over two and I'll write this third harmonic as two times ten meters over three this fourth harmonic is going to be two times ten meters over four this fifth harmonic could be written equivalently as two times ten meters over five since that's twenty fifths and now hopefully you see the pattern you realize okay I see what's going on here if I want the wavelength of the nth harmonic and could be like the first the second for the third so n is really just an integer one two three and so on I can figure out what that wavelength would be simply by taking two times the length of the string I'm going to write it as L so this applies to any string of any length as long as it's got nodes at the endpoints so take two times L and then just divide by n so in other words if I want the wavelength of this eighty-fourth harmonic I'll just take two times the length of my string and divide by 84 if I wanted the 30 third harmonic I take two times the length of the string over 33 and that would give me the wavelength of that harmonic now you should remember when we derived this equation we drew these pictures and these pictures all assume that the endpoints are nodes so this equation assumes you have a node node standing wave on a string which honestly is almost always the case since on all instruments with a string both ends are fixed so recapping when you confine a wave into a given region the wave will reflect off the boundaries and overlap with itself causing constructive and destructive interference for particular wavelengths you can set up a standing wave which means the wave just oscillates up and down instead of left to right in these standing waves the points where there's no motion are called nodes and the points of maximum displacement are called antinodes you can find the possible wave lengths of a standing wave on a string fixed to both ends by ensuring that the standing wave takes the shape of a simple harmonic wave and has nodes at both ends which if you do gives you a formula for the possible wave lengths for no nodes standing wave as being two times the length of the string divided by the number for the harmonic you're concerned with
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