Derivation of the 2D Wave Equation

hello everyone and welcome to another video today I'd like to continue our exploration of partial differential equations by revisiting the wave equation and this time looking at how it behaves if there are two spatial variables you might recall that some of our previous videos investigated deriving the one-dimensional wave equation we saw how this was a great way to model a system such as a vibrating string if you haven't already viewed it I would recommend stopping and watching that video first before continuing on with the discussion I will leave a link to that video in the description below so for now why don't we turn our attention to two-dimensional membranes such as something like a drum head for those of you who have watched some of my other videos I think you know that I'm not the best artist in the world so instead of trying to draw a diagram of what we're talking about today I thought it might be easier if I try to make a physical model of the system we're trying to work on so in this case the I'd like to take a membrane of some sort in this case I've you know quote-unquote borrowed one of my wife's towels here and you can see here is what I want to do here is introduce some tension to this and impose a boundary condition around some region on this membrane so to do that I'll also go ahead and use this springform pan that I also quote-unquote borrowed from the kitchen here and I'll just wrap this guy around it here and let's see if I can maybe secure the edges here with our with a rubber band here and what I'd like to do now is hopefully be able to introduce some tension on this membrane this is actually a lot harder than it looks to do without an assistant or without a flat table to work on but I think I can get this to work okay okay here we go okay alright so now you know this yeah this seems pretty reasonable so I'm trying to introduce a uniform tension along the face of this membrane here and we can see now that we've constrained the edge of this membrane to not move so now if I impose some kind of initial condition for example like flicking the back with my finger you can see this vibrate right and it wiggles I don't know can you see that maybe if I get a little bit closer maybe that might be easier if I if I flick this you see we get some vibrations in here I don't know that still seems a little bit difficult to July so so maybe here to help this to help us see the vibration here and to convince you that this surface is actually vibrating here I've also you know on my on my FEMA borrowing stuff from from people I've used some of these beads from my daughter's aren't kids so let me just sprinkle these around the surface now and what I'd like to do now is if we also now get this thing to vibrate you should be able to see these beads kind of won't crud whoops uh okay that that vibrated a lot more than I expected here and kind of made a giant mess oh well doesn't matter I've got I've got two small kids and a dog in the house so when somebody finds this mess in the morning I'm just gonna blame it on them but anyway I think that sets the stage here of what we'd like to do so let me see what we could do here is I would like to derive equations of motion for this system by considering a small area some da on this this membrane so you know what I really want to do is I kind of want to draw on this so we can see like a coordinate system but I wonder if my wife is gonna miss this thing hey hon I kind of want to draw on your towel let me know is that a problem okay I didn't hear anything did you um let's go ahead and do it anyway so let's proceed what I would like to do in this case now is let's let's attach a coordinate system to this membrane here so maybe I'll draw like let's make an X direction here in this this way I'll label this here is maybe X here and then we'll have a y coordinate in the opposite direction here and what I'd like to consider now is a small chunk of this membrane and we'll call this thing da so maybe I'll draw a little rectangle I guess it's pretty big here in this picture just to make it show up here but let's say that it has a width here of Delta X and it has a height here of Delta Y so this is this small little chunk of area that I'd like to consider here okay so what I like to do now is think about can we find a function here that describes how any bit of this membrane deflects in and out of the plane and any spatial coordinate x and y and at any temporal coordinate time T so that's the plan of attack why don't we go ahead and draw a diagram here on the whiteboard so we can add some details of what this little area da looks like all right so to get equations of motion let's first start off by writing a couple of assumptions for this problem here so the first assumption here is let's go ahead and assume that the density is uniform so in other words this is a something that's made out of the same material and the it's it's the same everywhere here so the density we're actually going to measure this in units of saying like kilograms per meter squared so it's so it's a mass per unit area here and we want to consider this being constant over the entire membrane okay so that seems pretty reasonable here um to here let's assume that the membrane is perfectly flexible flexible so membrane is perfectly flexible so in other words it has no resistance to bending okay so it's a thin membrane in that case this is not good for like a piece of sheet metal which definitely has some resistance to bending this is more like four pieces of thin rubber or latex or cloth or things like that right okay so assumption three here membrane is fixed along the boundary so this is not going to really play so much into the derivation of the partial differential equation for the system right now but it will definitely come into play when we actually try to solve this and apply some boundary conditions so let's consider a fixed boundary condition here and for here let's assume that also the tension is uniform and constant so the tension again let's use units here of force per unit length we're going to see in a second why we're using this measurement of tension here again this is not like a single string where yeah there's definitely a clear single value of tension throughout the string right this is now membrane where the tension is spread out over the entire surface so to make the unit scan and the math a little bit easier let's just conceptualize attention as force per unit meet per unit length here and we're gonna say that it's the same everywhere and it doesn't change even if the membrane is vibrating up and down here it's going to basically be the same tension everywhere okay and finally assumption number five here is that out of plane deflections are small right so really what we're looking for here in this case is actually this function u here which describes the outer plane the deflections of this membrane at any given X any given Y and at any time T here and we're going to assume that these are small relative to the geometry of the rest of the problem okay so with these assumptions in mind I think we can go ahead and draw a diagram of this system here so maybe let's do it like I want to try to make sure I use the the board space intelligently here let's draw it over here so what I'd like to do first here is let's just draw a diagram of this system kind of in the X Y direction here so we've got some kind of large membrane so like you know in this case it was our big dish towel here but what I would like to consider now is a small area of this dish towel that we'll call some rectangular chunk da okay and we said that all right the left side of this chunk here is at coordinate X and the right side of this chunk here is at coordinate X plus Delta X right if we consider this thing having a width of Delta X here and a height of Delta Y right this is where the left and right side of this thing are located we can do the similar for the bottom is located at Y and the top of this thing is located at Y plus Delta Y right all that seems pretty reasonable okay now you know what might be helpful is let's draw a larger version of of this picture here and we'll do this in three dimensions here so let me just draw a three dimensional coordinate system first here so here's X Y and then Z which is actually our deflection right out of the plane here so what I'd like to do is I want to get this cut this little chunk of surface out here from this top view and place it in this three dimensional view you know yeah you know what I actually might be helpful let's let's actually physically cut out this da out of our membrane right I've already this thing is already a goner here because I think I already drawn it with a sharpie so I think I'm gonna get in trouble anyway so if you're gonna if you're gonna get in trouble let's just let's just go the whole nine yard distance here so I'm just gonna cut out my little chunk of the membrane that I'm interested in and again this is difficult to do here boy I wish I had a table or something but I guess I might make it a little hard okay so let me cut this thing out almost there ah whoops all right it's falling apart all right tell you what switching switching tools let's just use scissors here I need my yeah I do need an assistant I should get my little daughter to do this she is like really good at cutting things out she's in kindergarten right now and I think in kindergarten all you do here is work on gluing and cutting things so she's an expert at this kind of stuff all right here we go I've cut out my little chunk da here right so again what I want to do is take this little section here overlay it in space here and then let's let's let's draw a rough picture of what this thing looks like right so let me grab a pen here maybe the easiest thing to do first is to draw it's projection into the XY plane here so maybe I'm gonna draw it like this here all right so here's its projection in the XY plane here right the left side of it again we said what's actually that should not go like that that should go like this right so here's X here's X plus Delta X here is y here is y plus Delta Y here right and now these points can get projected up to where the surface is actually located right so maybe we will draw it like something like okay something like something like that I think is a halfway reasonable drawing of this of this membrane here right okay so now what I'd like to consider here is let's consider what kind of forces are acting on this membrane here so along this front face here right that we've cut out there's going to be some force that if we look at sort of the average value here of where this is angled here then you're gonna have some force like this coming off of this face and the magnitude of that force here right isn't this going to be something like the T times the distance right so that will give us the units of Newton's or units of force so it's basically the tension times the distance or the length of this edge here right and we see that the length of that edge is just Delta X so this force here is T times Delta X similarly on the back side you're going to have another similar T times Delta X but it's going to be oriented and pointed in two slightly different direction because right we're taking this funny membrane here and this thing is laid out and it's draped over in in in a surface so the front face and the back face and the left side and the right side they're all pointing in sort of different directions and furthermore the slope of that pointing changes as X&Y change all over the face here right so we'll investigate that again in a second but okay so we get up these on the on the front in the back face similarly on the left and the right side we also have forces acting on this of T times Delta Y in this case and also T times Delta Y all right okay so it actually might behoove us now to look at a planar view here let's just look at this in the XZ plane so let's draw that a little more cleanly down here so if I draw this like such here you got X and Z which is really our deflection U of XY all right so in this case again let's use the same color here you got your membrane here like such here and this is now located here at X here and again this is located at X plus Delta X right and we got the force over here of magnitude T times Delta Y all right and over here you got T times Delta Y here right and again this picture is slightly misleading again here right I've drawn it like it's sort of making the assumption that you know again here's our little area Delta X Delta Y we're now looking at it directly kind of on an edge here and the way that picture is drawn it's basically it's misleading in the sense that I don't think I can get the camera angle right it's something so jeez yeah I can't it's something like like this here in that picture where it makes it look like there's no variation in the surface as Y moves as you go in and out of the board but one of these edges you know maybe that back edge is tipped up quite a bit so you definitely can see that there is some variation in slope as you move in the Y direction here but what we're drawing here on this board here is let's consider this to be sort of the average angle here of this membrane as you move along that direction so in other words what I'm going to assume here is let's call this angle here alpha and that is this average angle here of where this this average aggregate tensional force on that edge acts similarly on the other side we can do the similar thing here let's call this angle here beta for that average angle okay great so with this I think we're pretty good to go here let's go think about how can we go ahead and derive now some equations of motion based on this system right if we have a effectively a free body diagram of our system so let's just go ahead and apply Newton to this so first thing is let's let's consider this thing in the horizontal direction right so let's look at horizontal forces right so just looking at that free body diagram let's look at this in the x-direction here right so we basically have some of the forces in the horizontal I guess if you want to look at it well I get I guess what's a better notation for this because this is all sort of horses probably fine here horizontal right so in this picture down at the bottom here right we basically have something like what it's T Delta Y times Co sign of beta right minus T Delta Y cosine of alpha here right and due to our small angle approximation right we said that all right these small angles are basically 1 here so we end up with with 0 here right so this is great what this shows us here is there's no horizontal forces and in this situation due to our assumptions which basically means right no horizontal motion so the only motion this membrane is going to experience is in the vertical or the out-of-plane direction so let's look at that right now how about vertical forces all right so in the vertical forces right we can also do a similar thing some of the forces in the vertical direction right ok so just looking at that picture we got what T Delta Y sine alpha or no I'm sorry excuse me sine beta right sine beta minus T Delta Y sine alpha right so you got this force going up you got that force going down I just look at the vertical components here and now let's go ahead and use our small angle again approximation right which basically said if you had you know sine of some small angle let's call it theta here right didn't we say that's effectively theta here or the more useful way to look at this is this is effectively tangent of theta here right those are all the same thing because tangent is right it's just sine over cosine we said earlier cosine was just 1 so this is all approximately reasonable here right so let's plug in this expression for tangent into these over here right so what we end up with here is some of the forces in the vertical right we could just write this thing as T Delta Y times the quantity tangent beta minus tangent alpha all right ok let's look at what our tangent alpha our tangent beta and tangent alpha let's look at tangent beta here right tangent beta is if you come back over to this picture tangent is opposite over adjacent here right so it's almost like rise over run here which is effectively the physical slope of this system here at at the right edge in this case correct so what we see here is basically tangent beta is the same thing as partial U with respect to X at this location so maybe we should write that down here let's write this real quick here so we can say that tangent beta here right is basically partial U with respect to X evaluated at X is equal to X plus Delta X here right and at some Y value of again we got to be a little bit careful here as well actually no we don't have to be careful what I like to be kind of cautious about this here so this is at some again the Y location this can change here right so what we want to note here is let's just pick some y1 value which is in between y and y plus Delta Y here right because again all we're trying to do here is basically say that over here this was tangent of this sort of average aggregate angle beta here right well I want to equate that to the physical slope here but again we talked about earlier the physical slope of this thing in the X Direction is changing as Y goes in and out of the board right so I'm just gonna pick some value Y such that the right-hand side equals the left-hand side again it's if you if you want to just boil this down or reduce it's just like we're looking at averages here okay um okay so that's great we can do the exact same analysis for alpha here right this is d u DX so again tangent alpha if I look at this picture it is opposite over adjacent on this side or rise over run on this side or physical slope of U with respect to X here at the left side of the of the piece of a membrane here right so this is d u DX evaluated at X and y equals some other y2 value how about between y and y plus Delta Y right okay what's alternate notation for both of these we could also say that this here you know you could also write this as u X here at X plus Delta and at y1 right and similarly this thing is the shorthand notation would be UX at x and y 2 right awesome so let's substitute those in for this expression over here and what do we get here so we get oh and actually sorry ah vertical forms maybe we should have said this is some of the vertical on the left and right side right because we actually haven't dealt with the front and back we're going to deal with that in just a second here so really so we got tension Delta Y times UX of where was it yeah here X plus Delta X and y1 right all right - UX at x + y - right ok great this was some of the so some of the forces in the vertical on the left and right side okay you can do the exact same analysis for the front and the back face of this of this object here right of this small da and you would get virtually the identical expression here so let's just write that down what you would end up with you'd end up with some of the forces in the vertical on the front and back faces here it's gonna be tension Delta X here and now you would end up with uy at again some average value like x1 + y plus Delta Y here - uy - at Y great ok so the total sum of all of the forces on acting on that little object here is just the this one plus this one here so I think we're in a good spot to apply Newton here to look at the total motion of this system here right so noon second law is going to to govern this here so let's go ahead and write that down so Newton's second law right which was just mass is or sorry yeah or hell let's do this mass times acceleration is equal to the sum of all the forces right and if we're looking at this in vertical direction we can add little subscript fees here okay so alright what's an expression from mass so the mass of this thing is the density here right remember we said that this was units of kilograms per meter squared here right or a mass per unit area so I got to go ahead and take this density here multiply it by the area of that thing da well the area of this thing is just Delta x times Delta Y right great so here's our mass times acceleration in the vertical direction I think everyone would agree that that's just here's the deflection in the vertical direction or the position so if I take its second derivative with respect to time I get acceleration so here we go mass times acceleration is equal to this garbage down here so let's write that down here so it's it's T times well gosh I don't really want to rewrite it tell you what let's let's do this it is this garbage junk how about junk 1 plus junk - ok just don't want to write this all again here because I do I want to make a simplification here so now let's go ahead and and isolate the the u double dot here so I'll divide through by all of these gobbledygook numbers here right so we end up with the left-hand side is just D u squared DT squared is equal to this whole other side can basically reduce to tension over density times UX of X plus Delta X Y 1 minus UX X Y 2 all over Delta X here write this plus you get a very similar term uy X 1 y plus Delta Y minus uy X to Y all over Delta Y and let's close this square bracket to match that ok all right I want to look at this term right now look in fact let's look at the numerator here so the numerator here is what's this first term in the numerator this is the slope at X plus Delta X so this is effective we slope at the right side of the membrane - this here is slope at the left side of the membrane so the entire numerator it's basically the difference in slope or how much did the slope the X is the slope in the X with respect to X change so here's like a rise here right or how much did the slope change and here's your run as you move over a distance Delta X here right so if you think about this long enough this is rise over run it's basically asking how did the slope change as you move from Delta X here or it's basically saying this is you X X right similarly down here this term right here this is slope with respect to Y on the back face - slope with respect to Y on the front face so the numerator of this term is basically how much did the Y slope change over a run of Delta Y so this here is u YY so awesome this whole thing now can be rewritten in a nice compact form here of partial u squared with respect to t squared is equal to T over Rho times u xx + u YY right let's just call this thing this is some constant right it's just tension over over Rho this is like like a C squared here and if you look at this long enough right doesn't this look just like the it should be ringing some bells from our vector differential calculus discussion this is this is just laplacian of u so the nice compact format of this is basically partial square u with respect to t squared is equal to some constant c squared here times the laplacian of u and here you are here is our 2d wave equation right and in this case i again maybe we should we should fully write this out here so here C squared is tension over Rho it's just some number here associated with your problem here and and again if you look at this long enough if there was if if there was only one spatial dimension like if there was no Y if there if this was gone right we recover base tically are 1d wave equation so maybe the more efficient route to have this discussions we should have done this first and then simplified to the 1d wave equation that's the other way you could have a skin this cat here so now that we've got the actual here's our here's our 2d wave equation let's go ahead and think about solving this thing now in the in the next lecture all right so I think this is a good spot to stop so if you liked the video please subscribe to the channel it really helps me continue making these videos and I hope to see you at the next video when we will solve this for actually three independent variables right an X a Y and a time so I hope to catch you then bye

Channel: Christopher Lum

Views: 10,909

Rating: 4.9640718 out of 5

Keywords: 2D wave equation derivation, wave equation derivation, derive 2D wave equation, partial differential equation, wave equation PDE

Id: KAS7JBztw8E

Length: 27min 14sec (1634 seconds)

Published: Fri Nov 16 2018