 Continuing with our qualitative study of the wave equation, today we are going to introduce two properties which we may say are exclusive to hyperbolic equations and hence in the second order equation that we are going to study namely wave equation, heat equation, Laplace equation. So these properties are exclusive to wave equation and they are known as domain of dependence and domain of influence. We will discuss more on that in this lecture. So the outline for today's lecture is first we start with defining what is domain of dependence and domain of influence and then in short I write DOD and DOI. So we find out what they are for one dimensional wave equation and then what they are for 2 and 3 dimensional wave equations. So Cauchy problem for homogeneous wave equation is given by this the wave operator or dialomb version equal to 0 that means we are considering homogeneous wave equation and this is the Cauchy data ux0 is phi x utx0 equal to psi x. So we have already solved this Cauchy problem in earlier lectures. So there are two questions on interrelationship between the Cauchy data and solution. So we are going to pose those questions now. In 1 let you be a solution to the Cauchy problem for homogeneous wave equation. Let x0 t0 be a point in rd cross 0 infinity sometimes we say x0 t0 is a space time point, point in the space time. Of course u of x0 t0 being the value of the solution at the point x0 t0 the Cauchy problem it would depend on the Cauchy data that is not a surprise. So now the question is does the solution depend on the values of phi x and psi x at every x in rd that is the question. If the answer was yes or maybe then we will not be devoting a lecture to this topic therefore answer is not going to be all x in rd. So there will be a specific domain in rd we will soon see that. So set of all x in rd on which the solution u of x0 t0 depends through Cauchy data is called domain of dependence for the solution at the point x0 t0. Let us move on to the second question. Let you be a solution to the Cauchy problem for homogeneous wave equation. Let x0 be a point in rd. The Cauchy data at x0 recall the Cauchy data phi and psi are defined for x in rd. So the Cauchy data at x0 namely phi x0 and psi x0 is expected to influence the solution. Now does the Cauchy data at x0 influence the solution u of xt at every xt in rd cross 0 infinity? The answer is set of all xt in rd cross 0 infinity such that the solution u of xt is influenced by the Cauchy data at the point x0 in rd is called domain of influence or region of influence of the point x0. So here once again answer is going to be not every xt in rd cross 0 infinity the Cauchy data x0 is going to influence. Remark on the two questions and their answers. The two questions could have been asked for non-homogeneous wave equation also. Answer would still depend only on the solution the Cauchy problem for homogeneous wave equation. Why is that the Cauchy data and the source term do not interact? Recall the formulas that we have derived for the solution of Cauchy problem to a non-homogeneous wave equation. So past and future cause and affect. These are the two points of view that we can present domain of dependence and influence. Question one let us rephrase it. Suppose I am standing at a point x0 at time t0 that is at the point x0 t0 in the space time. What is causing or responsible for the current state u of x0 t0 from the past situation or data at time t equal to 0? Question two rephrased suppose I am standing at a point x0 in rd at the initial time t equal to 0. What are the points x t in the space time at which the data or situation at x0 at time t equal to 0 that is at the current time or initial time influences or affects the future state u of x t. So cause and affect past and future dependence and influence. Rest of this lecture is devoted to determine the domain of dependence and domain or region of influence for Cauchy problems for the wave equation in d space dimensions d equal to 1 to 3. The explicit formulae for solutions to Cauchy problems namely the Alambert formulae for d equal to 1, Fossum Kirchhoff formulae for d equal to 2 and 3 they will be used the formulae will be used. So let us move on to one dimensional wave equation and find out what is the domain of dependence. The Alambert formulae for the solution to Cauchy problem is given by u of x0 t0 equal to phi of x0 minus ct0 plus phi of x0 plus ct0 by 2 plus 1 by 2c integral x0 minus ct0 to x0 plus ct0 psi of s ds. So to compute the solution at the point x0 t0 what we need is the values of phi are needed exactly at two points x0 minus ct0 x0 plus ct0 and the value of psi are needed psi appears here it is an integral on the interval x0 minus ct0 to x0 plus ct0. Therefore the domain of dependence is the interval x0 minus ct0 comma x0 plus ct0 this is the interval which is the domain of dependence for the solution at the point x0 t0. So this is the picture here we have the point x0 t0 this is the interval on the x axis x0 minus ct0 x0 plus ct0. If you notice x0 minus ct0 is nothing but the line through x0 t0 the characteristic line given by x minus ct equal to x0 minus ct0 where it touches the x axis is precisely x0 minus ct0 comma 0 we are not writing that we just write x0 minus ct0. Similarly x0 plus ct0 is the point of intersection of this x axis and this characteristic x plus ct equal to x0 plus ct0. So this is the interval on the x axis on which the value of the solution at x0 t0 depends. So the solution at x0 t0 depends only on the Cauchy data from the interval x0 minus ct0 comma x0 plus ct0 what do we mean by this suppose I take two sets of Cauchy data phi psi and phi 1 psi 1 so that they agree on this interval x0 minus ct0 comma x0 plus ct0 that is phi is identically equal to phi 1 psi is identically equal to psi 1 on this interval. Let u1 u1 denote the solutions to the Cauchy problems for the wave equation with this Cauchy data phi psi and phi 1 psi 1 respectively then u of x0 t0 is equal to u1 of x0 t0. In fact u of xt is equal to u1 of xt for every xt in the triangular region determined by the two characteristic lines through the point x0 t0 on the x axis. For example, this is the point x0 t0 this is x0 minus ct0 this is x0 plus ct0 suppose I take a point which is inside somewhere here of course we know the value at this point will depend on this interval's value but on this interval phi and psi are equal therefore this holds for any arbitrary point that you take the solution will be the same with both the Cauchy data it is not going to change. So this is the domain of dependence picture once again x0 minus ct0 x0 plus ct0 so I have already demonstrated suppose I take a point here then the solution at that point depends on the values of the Cauchy data on this interval. So if you take a point here these are the characteristics passing through this point and where it touches on that is on this. So therefore if the Cauchy data phi psi and phi 1 psi 1 are coinciding on this interval then the solution will be same for both the Cauchy data that is u and u1 coincide on this triangular region. In particular changing the Cauchy data outside the interval x0 minus ct0 x0 plus ct0 has no effect on the solution at the point x0 t0 because the solution at x0 t0 depends only on the values of phi and psi in this interval therefore if you change it outside does not matter that was what was proved by considering the two Cauchy data that we considered phi psi and phi 1 psi 1 which are agreeing on the interval x0 minus ct0 x0 plus ct0 therefore solution is the same at x0 t0 outside this interval phi psi may not be same as phi 1 psi 1 that does not play any role at all. So that is the effect of change in initial data is not felt at the point x0 for all times t less than or equal to t0. Let us have a look at it again so this is the point x0 t0 what does it mean this is the point x0 this is x0 minus ct0 x0 plus ct0. Suppose I am standing at the point x0 this is a t direction right t direction suppose I am standing at some time t equal to capital T then I am at this point at this point solution is here right depends only on this interval. So up to this time it will depend only on the values here suppose you cross this time and stand here then yes this part will be new right this piece this this part will be new and here phi equal to phi 1 psi equal to psi 1 but here phi may not be equal to phi 1 may not be true okay. So therefore the solution at this point u of x let us call this point as x0, t1 so u of x0, t1 may not be same as u1 of x0, t1 because in this piece and in this piece phi and psi we have no information whether they coincide or not. So thus we may say that the solution at x0 t0 has a domain of dependence given by this interval. So let us look at the domain of influence of a point x0 on the x axis it is this set xt in r cross 0 infinity such that the domain of dependence of solution at xt contains the point x0. So what is that suppose this is my x this is my t directions suppose I am a point x0 now what is the domain of influence of x0 it contains those points xt such that the its domain of dependence contains the point x0. Let us consider a few points and see let us take a point p the solution at this point will depend on this interval and this is the domain of dependence for p it does not contain x0 therefore this p does not belong to domain of influence of x0 for example I am at this point in space time. Now the domain of dependence for this new point q is this interval and x0 falls inside that therefore q belongs to the domain of influence of x0. Let us consider one more point suppose I am here r the solution at this point this is the domain of dependence for r of course x0 is not there in it therefore r does not belong to domain of influence of x0. Suppose I take another point here at this point if you notice in the domain of dependence for this point yes x0 belongs to so therefore x0 influences the solution at s so s belongs to the domain of influence of x0. So since the domain of dependence of solution at xt is this interval x-ct x plus ct the domain of influence of x0 is this set let us look at this picture here okay if I take a point here of course x0 will lie in the domain of dependence for this point okay whereas if I take a point outside this v shape region definitely no okay. Similarly imagine this is r exactly same problem I have already written dots here the domain of dependence for r is this interval and x0 is not in that interval. So any point which is outside the v shape region x0 will not belong to the its domain of dependence and on the other hand any point inside this v in this region x0 will belong to the domain of dependence for solution at the points in this region domain of influence of an interval we have considered domain of dependence of a point now we are going to consider domain of influence for the interval how do we define that let us take an interval on the x-axis domain of influence of this interval should be the union of domain of influences of the points of the interval ab. Thus domain of influence of the interval ab turns out to be the set of all those points xt such that the domain of dependence of the solution at xt has a non-empty intersection with ab. So let us draw this line and take a piece here ab this is our interval now let us find out certain things for example I am at a point here okay. So at this point this is the domain of dependence for this point p and it does not intersect with ab so therefore p does not belong to domain of influence of the interval ab for example I take another point at this place let us call it q here this is the domain of dependence for q and it intersects this interval ab therefore the point q belongs to the domain of influence of the interval ab. Now the region which is given here is nothing but this this is x minus ct equal to b this is x plus ct equal to a. So if you take any point in this tub shaped region let us say here then definitely the domain of dependence will intersect ab for this point r and if you take a point here let us call it s then also it is going to intersect the interval ab the domain of dependence of that. So therefore the domain of influence of the interval is this particular set yeah. So here once again we have this picture if you take a point here this is the domain of dependence for p it is not intersecting ab if this is a point q the domain of dependence is here not intersecting on the other hand if you take a point here r then it is going to intersect it is much bigger than ab but definitely intersects ab and if I take a point here this is s then like that still intersecting. So if this is precisely the domain of influence of the region ab let phi psi b supported in this interval ab I am taking such a data that means phi and psi are 0 outside this interval ab. So this side phi and psi are 0 this side phi and psi are 0 for each fixed t positive where is the support of x going to u of x t. So let us fix time so this is the time right so this t equal to some capital time t. Now if you notice a point here ab will not influence this point p or any point which is to the left side of this particular line it will not influence therefore solution is 0 and similarly to the right side of this line to this side if you take any point q u at q is also 0 therefore only on this really it may be non-zero. So therefore the support is contained in this interval what is this point and what is this point that you can check it is going to be an interval actually this is a a has moved this side by time t we are going to see you can compute and see a minus ct similarly b has moved this side to this point this is b plus ct this is a minus ct the support will be contained in this. So support might increase of course you see original ab is the support for phi and psi and now the support is here for the solution at the time t equal to t it might increase but notice support is still compact. The support of a and support of phi and psi is inside ab it means it is a compact set support is a compact set now the support is inside this interval it means this function has compact support that is an interesting observation about the propagation of the initial disturbances. Let us look at the two-dimensional wave equation and domains of dependence and influence for them this is a formula for the Poisson kickoff formula for the solution of the Cauchy problem for the wave equation in 2D. Now if you notice the formula depends the values of phi and psi only on this disc therefore the domain of dependence for the solution at the space time point x1 x2 comma t is this disc of radius ct with center x1 x2 one may also consider closed discs that is not a problem because it is an integration right it is an integration so the boundary what is the difference between the closed disc and the open disc it is a boundary and that does not make any change to this integral it does not affect the integral so there is no plus here there is a problem okay fine okay so if you want it to be a closed set if you want it to be a closed set but there is no need for asking that so we do not take this point of view now what is the domain of influence that is given by set of all space time points such that distance between x1 x2 and y1 y2 is less than ct okay the y1 y2 x1 x2 distance is less than ct that is a domain of influence please convince yourself about this answer everything comes from this formula so this is a set of all those points which can be reached within time t from y what is this this is the distance between the boldface x and boldface y that is less than ct okay if you divide distance with c so distance by speed is less than this t so that means you reach within within the time t from y to x or x to y so three dimensional wave equation what are the domains of dependence and influence this is the formula Poisson Kirchhoff formula s of x comma ct is the sphere so therefore the domain of dependence is s of x ct because that is where the integrals are on right and domain of influence at the point y in rd is now norm x minus y equal to ct because of this sphere earlier it was a disc that is why it was less than t now this sphere therefore distance is precisely ct in other words those points which can be reached those points x which can be reached exactly at time t from the point y let us summarize we introduce the dual concepts of domains of dependence and influence like past and future like cause and effect extending the concepts of domains of dependence and influence for IVVP is straightforward domains of dependence and influence were computed explicitly explicit formula for the solutions were used in lecture 5.3 we will arrive at the same conclusions without using the explicit formula for solutions but of course you have to use something thank you.