 Continuing our qualitative analysis of the wave equation, today we are going to see what is called Huygens principle. The outline for today's lecture is, first we introduce what is called a strong form of Huygens principle and we show that it holds in 3D, a consequence sharp signal propagation in 3D and we give a small physical interpretation of the strong form of Huygens principle. Then we go on to introduce what is called weak form of Huygens principle and we show that it is satisfied in 2D and no sharp signal propagation in 2D and also in 1D and physical interpretation of the same and then we give a couple of remarks on Huygens principles. So Huygens principle is concerned with the propagation of information in space time. More precisely Huygens principle is concerned with solutions of the Cauchy problem for homogeneous wave equation, how the support of the Cauchy data propagates with time by the dynamics of the wave equation. An explanation on this restriction will be given later on in this lecture which is towards the end of this lecture. Huygens principle is stated in 2 different forms in the literature, they are strong form and weak form. The strong form of Huygens principle is commonly known as the Huygens principle. So when somebody is saying Huygens principle, it means it is a strong form of the Huygens principle. So when the propagation of waves is governed by homogeneous wave equation in D space dimensions where D is odd and D greater than or equal to 3, Huygens observed that if a wave is sharply localized at some time, then it will continue to be so for all later times. This observation does not hold when D equal to 1 or D is an even number. This observation is formulated as strong form of Huygens principle. We present Huygens principle from 2 dual points of view, namely in terms of domains of dependence and influence. Strong form of Huygens principle, the statement 0.1, the solution of Cauchy problem at a point x0, t depends only on the values of the Cauchy data on the sphere norm x minus x0 equal to ct. The values of Cauchy data at a point x1 influences the solution of the wave equation at all those points x, t for which norm x minus x1 equal to ct holds. Strong form of Huygens principle holds in 3 space dimensions. Recall the Poisson Kirchhoff formulae for the solution in 3D which is given by this formula where S of x, ct that is a domain of integration is a sphere with center at x having radius ct. This formula we have derived in lecture 4.5. Note that the solution at a point x0, t that is put x equal to x0. It depends on the values of phi and psi only on S of x0, ct that means those y which are a distance of ct from the point x0 when we are looking at u of x0, t. This is the statement 1 of the strong form of Huygens principle. Imagine that Cauchy data is concentrated at the point x1 in R3. This data will affect the solution at all those points x, t such that x1 belongs to the sphere S of x, ct which is a consequence of Poisson Kirchhoff formulae which is precisely the set set of all x, t in R3 cross 0 infinity such that norm x minus x1 equal to ct. What exactly is this set? Let us look at this quantity norm x minus x1 by c. In this norm x minus x1 is what? Distance between x and x1 and c is a speed. So distance by speed is time. So that is equal to t. So that is what happens for the points in this set. So this set consists of those points x, t such that x can be reached from x1 in time t with speed c that is the point x can be reached from the point x1 exactly in time t. Exactly means sharp as we normally use sharp in time t. Nothing less nothing more. This is the statement 2 of the strong form of Huygens principle. Let us turn our attention to sharp signal propagation. What does this mean? Consider the initial data phi n psi that is supported inside a ball of radius epsilon centered at a point x0, small ball. Poisson-Kirchhoff formula for 3D wave propagation suggests that the solution at the point x, t depends on the initial data only on the intersection of this b where the outside which cosh data is 0 and this sphere because the solution u of x, t is given in terms of integrals on the sphere. So therefore this sphere must intersect the support of phi n psi which is contained in b of 0 epsilon otherwise solution will be 0. Note that this sphere is expanding as t increases, as t increases sphere is expanding. As long as this intersection is empty solution will be 0. Since the sphere is expanding as time t increases with speed c there will be a time instant t e at which the intersection becomes non-empty. In fact t e can be given expression norm x minus x0 minus epsilon by c. So this is the ball where the support of the cosh data lies in b of x0, epsilon. So let me draw this line, this is the center x0. Let us call this point as x and this distance is epsilon. Therefore this distance is norm x minus x0 minus epsilon. This point lies on s t e. What is t e? Let us compute. So c t e is equal to norm x minus x0 minus epsilon. Therefore t e is equal to norm x minus x0 minus epsilon by c. Recall s t e is a sphere with center at x and the radius as c t e and it touches the support of 5 comma psi which is in the inside this ball x0 comma epsilon at this point. So therefore this point lies on s t e. So as you observe this sphere s t e which are centered at the point x and having radius c t as t increases t equal to t e is the precise time at which this sphere intersects b of x0 comma epsilon for the first time. Also a time t f after which the intersection will become m t. When does it become m t in this picture? After this. So after this let us call t let us say f for final. After this t f if you draw any other sphere for which t is bigger than t f it is going to be like that. So it does not intersect this ball. Thus the solution becomes 0 again after time t f. In fact t f equal to norm x minus x0 plus epsilon by c. That is an initial disturbance confined to a small ball of radius epsilon gives rise to an expanding spherical wave having a leading and a trailing edge and having support in an annular region of width 2 epsilon. 2 epsilon is precisely the diameter of the ball in which the Cauchy data is supported at each time instant t bigger than epsilon by c. In other words c t bigger than epsilon for such times this happens. This phenomenon is referred to as sharp signal propagation. So this is s t e and this is s t f. For t less than t e s t does not intersect the support b of x0 comma epsilon and for t bigger than t f once again s t does not intersect b of x0 comma epsilon. First let us understand what this c t means. c is the speed t is the time. So if you are moving at speed c you will travel a distance of c t in time t. So c t is the distance traveled in time t when you are moving with speed c. So recall that the Cauchy data is supported in b of x0 comma epsilon. So what does this sphere s of x0 comma c t plus epsilon represent? It consists of those points which can be reached from this ball in time t exactly at time t which can just touch here. Of course if they do all the points inside this even these points will also do. In time t they will travel maybe at this point it will reach. So this is the point which will reach precisely the other end point other end point. So if you take a point here in time t it would go away it will go out. So therefore solution will be 0 here. Let us call this as the center x0 this region if you draw a radius like this this width is actually 2 epsilon. So only if you are inside this annular region you may be non-zero. We would like to say it is non-zero but it could be 0 but definitely outside this u is 0. If you are here u is 0 if you are here u is 0 at time t that is support of x going to u of x comma t at a time t is inside the annular region of let us say width 2 epsilon. So interpretation of strong form of Huygens principle. Let us interpret the principle in terms of a physically relevant example. Assume that the wave equation model sound waves that propagate in our 3-dimensional world we can easily see that the strong form of Huygens principle holds in 3 dimensions. For example a sound wave generated by a speaker will reach the listener after some time depending on the distance from the speaker and of course at the speed which is the sound speed speed of sound in that medium. In fact the listener hears at the time instant t plus d by c the sounds produced by the speaker at the time instant t and d is the distance between them speaker and the listener. In other words the listener hears only silence then suddenly some speech is heard for a certain duration of time the time for which the speaker is producing sound and then suddenly once again silence which happens when the speaker pauses his speech or stops his speech. This illustrates a strong form of Huygens principle. Mathematically speaking we say that sound waves propagate exactly at the speed c. Note however that we hear a course and observe reverberation phenomena in enclosed spaces like caves. But this does not contradict the Poisson Kirchhoff formulae which were derived earlier they were derived for wave equation in the full space not with boundaries. This only means that wave equation is not suited to this situation we have to model differently not model it as wave equation on Rd or R3. Let us look at the weak form of Huygens principle. The solution of the Cauchy problem at the point x0 t depends only on the values of the Cauchy data up to this it is common with the strong form also in the ball in the strong form it was the sphere in the weak form we allow the ball norm x-x0 is less than or equal to ct that is a difference. The values of Cauchy data at a point x1 influences the solution of the wave equation at all those points x t for which norm x-x1 less than or equal to ct holds. So weak form of Huygens principle holds in two space dimensions these are Poisson Kirchhoff formulae including which if you want to compute the value of u at a point x, t x equal to x1, x2 here what you need is the values of phi and psi on this disc for all the y which belong to this that means what distance between y and x is less than or equal to ct less than ct but nothing wrong in saying less than or equal to ct also. So because integrals it does not matter what happens on the boundary. So this formula was derived in lecture 4.6 and as I have already pointed out if you want to find a solution at a point x0, t that is x1, x2 is now called x0 it depends only on the values of Cauchy data on this d of x0, ct yeah I should have used d because this is two dimension in any general dimension we use of course ball nothing wrong in using ball but it might confuse do not get confused in 2d ball means disc. So this is a closed disc center x0 radius ct this is the statement one or the weak form of Huygens principle. Imagine that the Cauchy data is concentrated at a point x1 in R2 this data will affect the solution at all those points xt such that x1 belongs to the closed disc dx, ct center x radius ct which is precisely the given by this set this is coming from Poisson Kirchhoff formula once again what exactly is this set let us look at norm x minus x1 by c norm x minus x1 is the distance between x and x1 c is a speed therefore what you get is time and time is less than or equal to t in this case it means this set consists of those xt such that x can be reached from x1 in a time which is less than or equal to t. Compare this with a situation in 3 dimensions where this time was exactly in time t now within time t that is the difference that is the point x can be reached from the point x1 within time t and this is the statement 2 of the weak form of Huygens principle no sharp signal propagation in 2d let the Cauchy data be supported inside a ball of radius epsilon centered x at the point x0 as before Poisson Kirchhoff formulae for 2d where propagation suggests that the solution at a point xt depends on the initial data only on the intersection of B this ball and this ball which is the domain of dependence for u of xt as in 3d wave propagation there will be a time instant at which the intersection becomes non-empty however there is no time after which the intersection will become empty. So let us illustrate in a picture I have demonstrated only st1 so let us draw the disc here because that is what comes in Poisson Kirchhoff formulae disc with center x let us call this point as x, x, radius is ct1 if you are writing this picture for st1 the disc is the inside part this disc clearly does not intersect the support. As t increases the disc size is increasing it is expanding this so let us look at this disc I am not able to draw the complete disc here because obvious reason picture is not fitting. Now this is let us call ste because this disc now of radius cte actually starts intersecting the support and hence it can start to pick up non-zero signals from the initial data. Now if you look at st2 corresponding thing will be let us call dt2 for simple notation dt2 is nothing but disc of radius ct2 centered at x. So that intersects because in this Poisson considerable portion it intersects. So therefore you expect non-zero solution and same is the case with the dt3 which corresponds to the boundary st3 till this time this is let us call is stf of course notation is a bad notation here I will explain why that is so this is a disc dtf goes. But then if t is bigger than tf which is depicted in the picture with the t4 the disc is going like that dt4 even dt4 intersects this support which is not a situation in 3 dimensions. In 3 dimensions stf is what that mattered and stf is the last time tf at which it intersects the support. For t bigger than tf st never intersects the support but in 2 dimensions it is not st it is important it is a dt which plays the role and dt continues to intersect the support. In fact if you take t bigger than tf the corresponding dt will always contain the support from this picture support of phi and psi. So therefore there is a time up to which the signal from the initial data is not received at this point x namely t after which it starts receiving after that it may continue to receive forever. Therefore there is a leading edge in this case but there is no trailing edge what we can expect is a kind of decaying trailing edge if you want we will discuss this later. So that is once an initial disturbance reaches at a point x not a time instant te it remains the effect will stay on forever unless the initial data have 0 averages this you ignore for the moment. This is also the case for 1D wave propagation we have already seen that. However the solution of a Cauchy problem to 2D wave equation decays it is true that it becomes non-zero but the effect slowly decays that means it goes to 0 slowly that is at each fixed x the value of uxt tends to 0 as t goes to infinity. This we will see in lecture 5.7 in 3D it was 0 after some time in 2D it is not going to become 0 after some time but it is going to decay as t goes to infinity which will be seen in lecture 5.7. So this phenomenon namely that of a slowly decaying trailing edge is known as diffusion of waves when d equal to 3 there is no diffusion as we saw that the solution become 0 after a time instant tf and then remain 0 for all t bigger than tf. In the case of 1D wave equation we saw this that there is no decay in the solution. In fact the solution is eventually constant when phi is identically equal to 0 and size of compact support with a non-zero average. Let us do the physical interpretation of weak form assume that the 2D wave equation models propagation of waves that are generated when a stone is thrown into a still pond of water. First observe circular waves propagating from the point where stone touched the water surface. Secondly we see that these circular waves propagate forever. We also observe that new circles are formed within the expanding circular waves which is a result of waves propagating at all speeds less than or equal to c. Thankfully we do not live in a one-dimensional or a two-dimensional world because if the propagation of sound waves is governed by 1D or 2D wave equation what will happen? The Alambert formula Poisson Kirchhoff formula imply that sound propagates at all speeds less than or equal to c that results in aqueous and phenomena of reverberation. So we should be happy that sound waves do not follow 1D, 2D wave propagation. Remark why compactly supported data? In the discussion of Huygens principles we considered propagation of compactly supported Cauchy data. The reasons are as follows what are the reasons? Recall that Huygens principle are concerned with propagation of supports of initial data with time. Thus considering arbitrary initial data is not meaningful. In fact we should ideally consider initial data that is supported at a single point like what is called a Dirac delta function. It is just at one point. To avoid technicalities that arise by considering such data we look at a good approximation of it namely that data is supported in a small ball. In other words a compact set. If the support is a small interval then it mimics the point supported data or a small ball then it mimics point supported data in higher dimensions. Now why no non-homogeneous equation? We consider only homogenous equations. Huygens principles are concerned with propagation of Cauchy data under the influence of the wave operator. One can also study the propagation effects of the wave operator in the presence of sources that is when the non-homogeneous wave equation is considered. One can study. For such situations there will be no special dimensions where there will be sharp signal propagation. Why? There will be waves travelling at all speeds as the sources are distributed all over the space time f of xt. Thus expecting any of the forms of the Huygens principles to hold in the context of non-homogeneous wave equations is unreasonable. So that is why we do not consider non-homogeneous equations. Let us summarize what we did in today's lecture. We can strong form of Huygens principle were introduced. If strong form of Huygens principle holds then so will be the weak form. Each form of Huygens principle is stated with two points of view. Statement one represents the domain of dependence point of view. Statement two represents the domain of influence point of view. Huygens principles are better understood from the domain of influence point of view. The minimum number of space dimension that allows sharp signal propagation is 3. We have observed this. Thank you.