[Music] so far what we've considered are waves that move along and as they do they transfer energy and because they're transferring energy we can say that these waves are progressive as they progress along but what I'd like to look at now are another type of wave and this kind of wave it doesn't transfer energy but it stores it and what we're going to look at instead waves that uh don't seem to move at all and these are called standing or stationary waves so it's these standing or stationary waves where we store energy and they're quite different to the progressive waves that you've been used to so far so how are these waves formed well if we have a progressive wave uh that is moving along and it reflects off some kind of boundary or perhaps an end point uh the reflected wave will come back and this time the reflected wave will have the same kind of amplitude and phase difference and it's the interaction of these two waves uh where we look at superp position and the overall resultant that gives us a standing wave what I'm going to look at here is a series of standing waves that we can have set up perhaps on a string and provided we have a fixed end at both sides the wave that we set up can't move up and down at the ends cuz it's in a fixed position and what I have here is the first standing wave that we produce and what we have is at the end it can't move and this is what we call a node or effectively there's no displacement at the node we also have a point of Maximum displacement which is an opposite to a node and this is what we call an anti node and all I've drawn in here is fact the maximum and the minimum position of perhaps a string which is vibrating I can repeat it for perhaps the second uh kind of standing wave that we can fit in so again we're going to have the ends must be constrained but this time there's another wave that is actually the length of one complete wavelength and if I draw that as one of its uh its waveforms the opposite to this at The Other Extreme half a cycle later looks a bit like this and again we have a node at each end we also have a node or a point of node displacement in the middle and then we have an anti node and another antinode now the key thing about this is that because this is one wavelength the distance from a node to another antinode is equal to the wavelength of that wave over two and this one here this is also equal to the wavelength over two uh and and so on and if we repeat the process what we have here although slightly dodgy drawn we do have uh various other standing waves that can be set up uh within the same distance from the diagram so far they all look a bit static so this is what it would look like here we have a couple of nodes and the particles moving d and down in the middle and then we also have another wave that has three nodes and also one that has four looking in a bit more detail at these various diagrams here perhaps we have a string that has length L and what we have here is the first standing wave that we can produce what we call this is the fundamental for this fundamental because the length is equal to the wavelength over two we can say that the wavelength is equal to 2 L and this happens at a certain frequency which we call F or F0 and that is the fundamental frequency of that standing wave now there's another one here this time what we call this is the second harmonic the reason being that the fundamental frequency is the first harmonic and the second harmonic the wavelength is equal to uh the length of that string and we call this uh where basically has twice the frequency of the first one and what we then have is the third and the fourth and what we find of many musical instruments uh is that uh because we have all these harmonics we have various multiples of that fundamental frequency so what we find is that there are a number of modes of vibration various harmonics uh and we still have the nodes and antinodes uh and as we have a higher and higher frequency we can see the wavelength gets smaller and smaller