Question Let be topological spaces. Let be a bounded, continuous function. Is the function continuous?

Intuitively, one may be tempted to say “yes”. Indeed, there are plenty of examples where the answer is in the positive. The simplest one is when we can replace the infimum with the minimum:

Example Let the space be a finite set with the discrete topology. Then is continuous.
Proof left as exercise.

But in fact, the answer to the question is “No”. Here’s a counterexample:

Example Let with the standard topology. Define

which is clearly continuous. But the infimum function is roughly the Heaviside function: if , and if .

So what is it about the first example that makes the argument work? What is the different between the minimum and the infimum? A naive guess maybe that in the finite case, we are taking a minimum, and therefore the infimum is attained. This guess is not unreasonable: there are a lot of arguments in analysis where when the infimum can be assumed to be attained, the problem becomes a lot easier (when we are then allowed to deal with a minimizer instead of a minimizing sequence). But sadly that is not (entirely) the case here: for every , we can certainly find a such that . So attaining the infimum point-wise is not enough.

What we need, here, is compactness. In fact, we have the following

Theorem If are topological spaces and is compact. Then for any continuous , the function is well-defined and continuous.

Proof usually proceeds in three parts. That follows from the fact that for any fixed , is a continuous function defined on a compact space, and hence is bounded (in fact the infimum is attained). Then using that the sets and form a subbase for the topology of , it suffices to check that and are open.

Let be the canonical projection , which we recall is continuous and open. It is easy to see that . So continuity of implies that this set is open. (Note that this part does not depend on compactness of . In fact, a minor modification of this proof shows that for any family of upper semicontinuous functions , the pointwise infimum is also upper semicontinuous, a fact that is very useful in convex analysis. And indeed, the counterexample function given above is upper semicontinuous.)

It is in this last part, showing that is open, that compactness is crucially used. Observe that . In other words an open set. This in particular implies that there exists a “box” neighborhood contained in . Now using compactness of , a finite subset of all these boxes cover . And in particular we have

and hence is open. Q.E.D.

One question we may ask is how sharp is the requirement that is compact. As with most things in topology, counterexamples abound.

Example Let be any uncountably infinite set equipped with the co-countable topology. That is, the collection of open subsets are precisely the empty set and all subsets whose complement is countable. The two interesting properties of this topology are (a) is not compact and (b) is hyperconnected. (a) is easy to see: let be some countably infinite subset of . For each let . This forms an open cover with not finite sub-cover. Hyperconnected spaces are, roughly speaking, spaces in which all open nonempty sets are “large”, in the sense that they mutually overlap a lot. In particular, a continuous map from a hyperconnected space to a Hausdorff space must be constant. In our case we can see this directly: suppose is a continuous map. Fix . Let be open neighborhoods of . Since is continuous, is open and non-empty (by the co-countable assumption). Therefore for any pairs of neighborhoods. Since is Hausdorff, this forces to be the constant map. This implies that for any topological space , a continuous function is constant along , and hence for any , we have is continuous.

One can try to introduce various regularity/separation assumptions on the spaces to see at what level compactness becomes a crucial requirement. As an analyst, however, I really only care about topological manifolds. In which case the second counterexample up top can be readily used. We can slightly weaken the assumptions and still prove the following partial converse in essentially the same way.

Theorem Let be Tychonoff, connected, and first countable, such that contains a non-trivial open subset whose closure is not the entire space; and let be paracompact, Lindelof. Then if is noncompact, there exists a continuous function such that is not continuous.

Remark Connected (nontrivial) topological manifolds automatically satisfy the conditions on and except for non-compactness. The conditions given are not necessary for the theorem to hold; but they more or less capture the topological properties used in the construction of the second counterexample above.

Remark If is such that every open set’s closure is the entire space, we must have that it is hyperconnected (let be a closed set. Suppose is another closed set such that . Then and vice versa, but is open, so . Hence cannot be written as the union of two proper closed subsets). And if it is Tychonoff, then is either the empty-set or the one-point set.

Lemma For a paracompact Lindelof space that is noncompact, there exists a countably infinite open cover and a sequence of points such that if .

Proof: By noncompactness, there exists an open cover that is infinite. By Lindelof, this open cover can be assumed to be countable, which we enumerate by and assume WLOG that . Define and inductively by: and choose .

Proof of theorem: We first construct a sequence of continuous functions on . Let be a non-empty open set such that its closure-complement is a non-empty open set ( exists by assumption). By connectedness , so we can pick in the intersection. Let be a sequence of points converging to , which exists by first countability. Using Tychonoff, we can get a sequence of continuous functions on such that and .

On , choose an open cover and points per the previous Lemma. By paracompactness we have a partition of unity subordinate to , and by the conclusion of the Lemma we have that . Now we define the function

which is continuous, and such that . But by construction , which combined with the fact that shows the desired result. q.e.d.

Question
What is the meaning of gauge invariance in the jet-bundle treatment of partial differential equations?

The answer, actually, is quite simple.

Review of geometric formulation PDE
We consider here abstract PDEs formulated geometrically. All objects considered will be smooth. For more about the formal framework presented here, a good reference is H. Goldschmidt, “Integrability criteria for systems of nonlinear partial differential equations”, JDG (1967) 1:269–307.

A quick review: the background manifold is assumed (here we take a slightly more restrictive point of view) to be a connected smooth manifold. The configuration space is defined to be a fibred manifold . By we refer to the fibred manifold of -jets of , whose projection where for we use for the canonical projection.

A field is a (smooth) section . A simple example that capture most of the usual cases: if we are studying mappings between manifolds , then we take the trivial fibre bundle. The -jet operator naturally sends .

A partial differential equation of order is defined to be a fibred submanifold . A field is said to solve the PDE if .

In the usual case of systems of PDEs on Euclidean space, is taken to be and the trivial vector bundle. A system of PDEs of order is usually taken to be where

is some function. We note that the domain of can be identified in this case with , We can then extend to a fibre bundle morphism.

If we assume that has constant rank, then is a fibred submanifold of , and this is our differential equation.

Gauge invariance
In this frame work, the gauge invariance of a partial differential equation relative to certain symmetry groups can be captured by requiring be an invariant submanifold.

More precisely, we take

Definition
A symmetry/gauge group is a subgroup of , with the property that for any , there exists a with .

It is important we are looking at the diffeomorphism group for , not . In general diffeomorphisms of will not preserve holonomy for sections of the form , a condition that is essential for solving PDEs. The condition that the symmetry operation “commutes with projections” is to ensure that , which in particular guarantees that extends to a diffeomorphism of with itself that commutes with projections.

From this point of view, a (system of) partial differential equation(s) is said to be -invariant if for every , we have .

We give two examples showing that this description agrees with the classical notions.

Gauge theory. In classical gauged theories, the configuration space is a fibre bundle with structure group which acts on the fibres. A section of induces a diffeomorphism of by fibre-wise action. In fact, the gauge symmetry is a fibre bundle morphism (fixes the base points).

General relativity. In general relativity, the configuration space is the space of Lorentzian metrics. So the background manifold is the space-time . And the configuration space is the open submanifold of given by non-degenerate symmetric bilinear forms with signature (-+++). A diffeomorphism induces and hence a configuration space diffeomorphism that commutes with projection. It is in this sense that Einstein’s equations are diffeomorphism invariant.

Notice of course, this formulation does not contain the “physical” distinction between global and local gauge transformations. For example, for a linear PDE (so is a vector bundle and is closed under linear operations), the trivial “global scaling” of a solution is considered in this frame work a gauge symmetry, though it is generally ignored in physics.

Suppose the repository on the old server is called “WorkSVN” as in my previous post. First we need to dump the contents from the old server by

svnadmin dump /home/XXYY/WorkSVN > WorkSVN.dmp

We then copy the file over to the new server. On the new server issue

svnadmin create ~/WorkSVN

as before, and followed by

svnadmin load ~/WorkSVN< WorkSVN.dmp

to load the database dump.

Then create the working copy as usual

svn co file:///home/XXYY/WorkSVN/PaperDrafts ~/PaperDrafts

For the client computers, update the server path via

svn switch –relocate OLDURL NEWURL

You can find the oldurl by reading “svn info” of the working copy.

In section two of the paper, Sasaki consider “extended transformations and extended tensors”. Basically he wanted to give a way to “lift” tensor fields from a manifold to tensor fields of the same rank on its tangent bundle. And he did so in the language of coordinate changes, which geometrical content is a bit hard to parse. I’ll discuss his construction in a bit. But first I’ll talk about something different.

The trivial lifts
Let be smooth manifolds, and let a submersion. Then we can trivially lift covariant objects on to equivalent objects on by the pull-back operation. To define the pull-back, we start with a covariant tensor field , and set by the formula:

where the , and we use that . Observe that for a function , the pull-back is simply .

On the other hand, for contravariant tensor fields, the pull-back is not uniquely defined: using that is a submersion, we have that , so while, given a vector field on , we can always find a vector field on such that , the vector field is only unique up to an addition of a vector field that lies in the kernel of . If, however, that is Riemannian, then we can take the orthogonal decomposition of into the kernel and its complement, thereby getting a well-defined lift of the vector field (in other words, by exploiting the identification between the tangent and cotangent spaces).

Remarkably, the extensions defined by Sasaki is not this one.

(Let me just add a remark here: given two manifolds, once one obtain a well defined way of lifting vectors, covectors, and functions from one to the other, such that they are compatible (), one can extend this mapping to arbitrary tensor fields.)

The extensions defined by Sasaki
As seen above, if we just rely on the canonical submersion , we cannot generally extend vector fields. Sasaki’s construction, however, strongly exploits the fact that is the tangent bundle of .

We start by looking at the vector field extension defined by equation (2.6) of the linked paper. We first observe that a vector field on a manifold is a section of the tangent bundle. That is, is a map such that the composition with the canonical projection is the identity map. This implies, using the chain rule, that the map is also the identity map. Now, is the projection induced by the projection map , which is different from the canonical projection from the tangent bundle of a manifold to the manifold itself. However, a Proposition of Kobayashi (see “Theory of Connections” (1957), Proposition 1.4), shows that there exists an automorphism such that and . So as a differential mapping induces a map , which is a map from the tangent bundle to the double tangent bundle , which when composed with the canonical projection is the identity. In other words, is a vector field on .

Next we look at the definition (2.7) for one-forms. Give a one-form on , it induces naturally a scalar function on : for , we have taking value . Hence its differential is a one-form over .

Now, what about scalar functions? Let be a one-form and be a vector field on , we consider the pairing of their extensions to . It is not too hard to check that the corresponding scalar field to , when evaluated at , is in fact , the derivative of the scalar function in the direction of at point . In general, the compatible lift of scalar fields to is the function .

Using this we can extend the construction to arbitrary tensor fields, and a simple computation yields that this construction is in fact identical, for rank-2 tensors, to the expressions given in (2.8), (2.9), and (2.10) in the paper.

The second extension
The above extension is not the only map sending vectors on to vectors on . In the statement of Lemmas 3 there is also another construction. Given a vector field , it induces a one parameter family of diffeomorphisms on via that maps . Its differential is a vector field over .

The construction in the statement of Lemma 4 is the trivial one mentioned at the start of this post.

For convenience, the space-time we construct will be spherically symmetric, and we will only consider spherically symmetric solutions of the wave equation on it. We will also focus on the 1+3 dimensional case.

Spherically symmetric space-times and their wave equations

In a 1+3 dimensional spherically symmetric space-time, associated to each point (unless the point is on the symmetry axis; that is, a fixed point under the symmetry action) is a two-dimensional sphere, the orbit of the point under the symmetry. Much like the way we define spherical coordinates for 3-dimensional Euclidean space, we can define an analogous system of coordinates for such manifolds. (For more detailed and more involved discussion of this, see Szenthe, On the global geometry of spherically symmetric space-times, Math. Proc. Cam. Phil. Soc. (2004) 137:741–754.)

The spherical symmetry in particular implies that, away from the symmetry axis, our space-time take the form of a warped product where is the space-time, is the axis, is the two-dimensional quotient manifold, is given the standard metric on the unit sphere, and the warping function is the area-radius of the symmetry spheres. (That is, is defined to be the square root of times the surface area of the orbit of the spherical symmetry.) So we can build a coordinate system on by taking an arbitrary coordinate system on the 1+1 dimensional Lorentzian manifold and joining to it some coordinate system on the standard sphere.

Now, any 1+1 Lorentzian manifold admits a (in fact a large number of) double null coordinate system. That is, we can always find a pair of functions so that they are linearly independent functions, with under the induced Lorentzian metric on (this follows from the fact that at each point , the tangent space admits two such preferred directions, and that Frobenius theorem is trivial in two dimensions). Under an assumption of time-orientability, we can find some function such that the metric on can be expressed as

Equation 27

where is the standard metric on the sphere.

Let us now consider a spherically symmetric solution to the wave equation on such a space-time.

Equation 28

Expanding the d’Alembertian operator as the Laplace-Beltrami operator for the Lorentzian manifold , we get that

Equation 29

which in particular implies that the function will solve the following equation

Equation 30

So in particular will solve a 1+1 dimensional wave equation with source term on manifold (which we can partially complete by adding the axis and imposing the Dirichlet boundary condition there). This will be the starting point of our construction (recall the support propagation properties of the wave equation from Part IIb of this series).

A “radiating” space-time
Consider the following partial Penrose diagram

Figure 5

To prescribe the geometry we do so by describing the functions as functions of the coordinates; we will build the space-time such that in the yellow regions of Figure 5 the space-time is Minkowskian, and that we have a perturbation in the red region. The red region we will assume to be bounded by and . This we will take to be a poor-man’s version of a radiating space-time, where the red region represents some sort of out-going gravitational disturbance. Of course, here we prescribe the metric freely, rather than try to solve for it using some sort of the dynamical model, so the actual form of the metric is somewhat aphysical. Nevertheless this should give some indication of why it is not feasible to expect too good a decay of the solutions of linear wave equations on arbitrary space-times.

On the yellow region near future time-like infinity, we will insert in the Minkowski space solution in the usual parametrization. That is, we assume that there exists some such that , and that the function identically.

In the red region, we choose the function so that it joins smoothly to the first yellow region where . To be more precise, will be taken to be equal to satisfy with and . By prescribing initial conditions on the bottom left boundary of the figure, we can assume that in the region we have . On the other hand, we necessarily have that there is a function of alone. By assuming that are sufficiently small, we can guarantee that in the region, and so we can define a change of variables so that in the coordinate system, we have that . We can then set in the region so that , thereby enforcing that the region is also Minkowskian. Lastly we take an arbitrary positive, smooth choice of in the red region to join the values we have now prescribed for the two yellow regions.

Tail for homogeneous linear wave equation

On this space-time we just constructed, let us consider an initial data for the wave equation such that is an incoming wave-packet. That is, we can assume that for , we have , and that , and . Such solutions are fairly generic in the space of compactly supported Cauchy data. Then using the fact that , we can integrate Equation 30 to obtain, inductively, that for . Furthermore, integrating the expression for , we have that in the red region, which implies, for sufficiently small, that in the red region.

Therefore the coefficient on the right hand side of Equation 30 decays like , so is in particular integrable in the red region. So for small enough , we can iterate the fundamental solution representation for the 1+1 dimensional wave equation to get a convergent series representation of the solution , with all terms non-negative. The first term of the iteration guarantees that is bounded below inside for any fixed . Which then implies that is bounded below by when is sufficiently large along .

Propagating forward into the yellow region, we have that in the yellow region you have decaying like . We can transfer this to an estimate on : in the yellow region , we have using that vanishes on the center axis. But is conserved along , so we actually have that . Dividing through by we get a lower bound that, within the eventual Minkowski region, the solution to the original linear, homogeneous wave equation exhibits a tail.

A more physical decay rate

[Thanks to Mihalis for reminding me of this!] One sees from the argument above that the tail decay rate is intimately tied to the rate of decay of in the wave region. And indeed, we can run through the exact same argument as above with the rates

for any power . This will lead to the appearance of a corresponding tail of size

.

The above construction shows that in arbitrary Lorentzian manifolds, one can not do better than a decay. But in the context of general relativity, this heuristic leads to being the more physical decay rate. This has to do with the fact that for physical space-times appearing in relativity theory, we expect the space-time to have a well-defined Bondi mass (which is the limit of the Hawking mass at null infinity. Formally we can read off the Bondi mass from the term in the asymptotic expansion of . For the choice of given above, with the Bondi mass blows up at infinity (the asymptotic expansion is not well defined). For , the Bondi mass is zero. So heuristically we expect the case to correspond to the physically interesting case where gravitation has non-trivial long range effects. And for this rate we have a cubic decay law expected (as consistent with the so-called “Price’s Law” in general relativity).

In this particular case we will consider the following system of wave equations on 1+3 dimensional Minkowski space, with the assumption of spherical symmetry in force.

Equation 21

The system as a whole, as an equation on the vector is nonlinear, since the right hand side depends nonlinearly on the vector itself. However, we recognize that taken individually the components of the vector decouple, and each component solves a (possibly inhomogeneous) wave equation with no nonlinearities. (So we cheat a little bit. But this is how we can make the successive approximation scheme converge after finitely many steps.)

Recall that under the spherical symmetry change of coordinates, writing again and , we can rewrite the equations as

Equation 21′

Why this particular form on nonlinearities? It is known that generic nonlinearities, quadratic in , lead to formation of singularities in finite time, with the number of spatial dimensions is less than or equal to three. In dimension 1, we can explicitly illustrate this: consider the wave equation . Then along the curves of constant , the quantity verifies an equation of the form , whose solution is . So if is negative at any point when , the solution will develop a singularity in finite time.

In low spatial dimensions, the rates of dispersion are slower (the energy density is expected to drop as the inverse of the area of the wave front, which in dimensions grows like ), and compared to the positive feedback coming from a generic quadratic derivative nonlinearity, may not be enough to radiate away the self-interaction sufficiently fast. This motivated my thesis advisor, Sergiu Klainerman, to study, when he was doing his own PhD, the conditions sufficient to guarantee the quadratic nonlinearities are not too strong. This lead to the recognition that the so-called null structures play an important role in the study of wave equations in low spatial dimensions. To cut the story short, the form chosen for the nonlinearity above renders it a null form, so it should be better behaved (decay faster) than the most general quadratic type nonlinearities. By choosing this example, my goal is to illustrate the fact that even in the good scenarios, one expect to have some fairly strong nonlinear tail effects. (A detailed discussion of this topic would take us too far from the focus of this post. Perhaps I’ll write about it in a part IV to this series.) (That we need a and a is also related. If we only have the function and choose the nonlinearity for to be , we see that the null form cancellation is too strong, and we won’t be able to see the effect at the level of the first iterate.)

Anyway, back to the example.

Figure 4 (again)

The key observation is captured already in Figure 4 (which we recall again here above): when the red dot is “sufficiently far in the future”, the crosshatched region only intersects the “outgoing legs” of the cyan region (using the fact the incoming wave, after bouncing off the center axis, becomes outgoing also). In other words, where the crosshatched region and the cyan region intersects, the wave is purely outgoing.

Now suppose the cyan region is a representation of the support of and , and we are trying to evaluate at the red dot (the part coming from initial data can be neglected due to linearity). To do so we integrate in the intersection of the crosshatched and cyan regions. Now in that region, Equation 21′ implies that and are outgoing waves, that is there derivatives vanish. Furthermore the wave equation also means their derivatives are constant along lines of constant .

Equation 22

Using the constancy of along constant lines, we have that , and similarly . So we get that

Equation 23

Now, the area integrated over is determined both by the cyan and crosshatched regions. The cyan region coming from determins the limits of the integration, while the crosshatched region determines the limits. We will just assume that the limits are some fixed values and , which is completely determined by the compact support of the initial data. The limits, however, depends on where we will evaluate . If we choose to evaluate at point , we see that by the requirement on the central axis, the lower bound of the region is precisely , and the upperbound . Therefore we have that, for the red dot sufficiently far in the future where the picture in Figure 4 is a good representation,

Equation 24

Now, sufficiently far into the future, is sufficiently large, and hence is very large. This means that we can throw away the second term in the parentheses as negligible. For generic and (note that if there is actually a cancellation so that the contribution from the nonlinearity vanishes), we can find a lowerbound once become large enough (the upper bound is trivial)

Equation 25

And in Equation 25 we capture the behaviour of the nonlinear tail. For any fixed , the integral . So we have that

Equation 26a
for fixed

giving us the “interior decay” rate. On the otherhand, if we were to take the limit toward with fixed, we have that the total integral is controlled by , and so we have the “radiative decay” rate coming purely from the weight:

Equation 26b
for any sufficiently large fixed .

In both Equations 26a and 26b, the “eventually vanishing” phenomena observed in the linear case for data with compact support are destroyed.

Lastly, just note that Equations 26a,b can be glued together with boundedness on bounded regions to get the global estimate

which captures a more generic (in the sense of stability under nonlinear perturbations) behaviour of spherically symmetric waves on 1+3 dimensional Minkowski space.

Duhamel’s Principle

To study how one can incorporate inhomogeneous terms into a linear equation, and to get a qualitative grasp of how the source term contributes to the solution, we need to discuss the abstract method known as Duhamel’s Principle. We start by illustrating this for a very simple ordinary differential equation.

Consider the ODE satisfied by a scalar function :

Equation 13

when , we can easily solve the equation with integration factors

Using this as a sort of an ansatz, we can solve the inhomogeneous equation as follows. For convenience we denote by the anti-derivative of . Then multiplying Equation 13 through by , we have that

Equation 14

which we solve by integrating

Equation 15

If we write , then we can rewrite Equation 15 as given by an integral operator

Equation 15′

So what does Equations 15 and 15′ mean? The term by itself would indicate a solution to the homogeneous version of ODE 13, evaluated at time , with initial data prescribed at time as . So in other words, the total solution of the ODE 13 at time corresponds to the “free evolution” of the initial data prescribed at time , plus an infinitesimal part for each time instant between corresponding to the "free evolution" of initial data from time to .

Qualitatively, what is going on is that at every time instant, the "source term" gives a small kick to the solution. Because of the linearity of the corresponding homogeneous problem, each small kick evolves independently of the free evolution, and also independently of other small kicks coming from other instants in time. At the end to evaluate the solution at time , all one needs to do is to add up the contribution from the sum total of these small kicks, and add to it the free evolution of the homogeneous problem given by the initial data.

Now, recall how many statements about scalar ODEs can be transferred to vector valued ODEs: we can treat to be vector valued in Equation 13, with to be matrix valued. The defining the usual way for matrices, we get that Equations 15 and 15' equally well apply in this context.

Similarly, with a bit more care, we can generalise this to infinite dimensional systems.

For the wave equation, observe that we can write it as a first order ODE on some Hilbert space: let , then the wave equation can be written as

Equation 16

so at least on a formal level (which can in fact be made rigorous) we can obtain the solution for the inhomogeneous wave equation by considering the solutions for the homogeneous version.

To recapitulate, and to introduce some terminologies, equations like Equation 13 above, where are “time dependent” functions that are scalar-, vector-, or Hilbert-space-valued, and where is a “time dependent” family of scalar-multipliers, linear transformations, or linear operators (in the three cases mentioned) are first order evolution equations. The corresponding object (which is now is a two-parameter, depending on initial and final times, family of multipliers, linear transformations, or linear operators) goes under the names of free propagator, Green’s function, or fundamental solution, as it denotes the (linear) change for solutions of the homogeneous problem between initial time and final time .

As an aside: Interesting to note is that each evolution equation corresponds to a category. The objects are sets representing the time-dependent spaces of data. The morphisms are precisely that maps between the different times. Well-posedness of the initial value problem relative to one element in the class of objects is then the statement that is an initial object (existence) and that for any two objects the class of morphisms consists of at most one arrow (uniqueness). Time reversibility will then be the statement our category is in fact a groupoid, and hence that is also a terminal object. Linearity amounts to requiring the objects be vector spaces and the arrows be vector space homomorphisms. Time translation symmetry would be the existence of a functor collapsing our category to a monoid. (At least, if I am not mistaken….)

Example: 1D wave equation

Let us illustrate this by considering the wave equation in 1 spatial dimension. (Which, again, using the reflection trick described in the last post, gives us a solution to the case with Dirichlet boundary conditions, and also then to the spherically symmetric wave equation in 3 spatial dimensions.) We first try to write down the free propagator in terms of the first-order formalism above. Recall Equation 7 for the solution to the homogeneous problem with initial data

Equation 17a

Now observe that also solves a homogeneous wave equation by linearity, and using that for a solution , we have that the initial data for is . Hence we also have the expression

Equation 17b

The time translation symmetry tells us that the propagator . And Equations 17a,b tell us what the map looks like. Now, going back up to Equation 16, we see that the vector representation of only contributes to the second slot. So we have

Equation 18

and thus, the source part of the full solution to the inhomogeneous problem, would be

Equation 19

which perhaps is easier seen with a picture.

Figure 2

In Figure 2, we assume that we want to consider the initial value problem for Equation 16 (of equivalently, the inhomogeneous wave equation) in spatial dimension 1. Let us assume that the initial data is prescribed on the axis, and the initial data is identically zero. (In other words, we just want to consider purely the contribution coming from the source term, by setting the first term on the right hand side of Equation 15′ to be 0. In the picture, the green region represents the support of the source function . Suppose we want to evaluate the solution at the red dot, then Equation 19 tells us that is given by the total integral of in the blue, crosshatched region, which is bounded below by the initial time slice, and above by the null lines emanating from the red dot.

(In general, given a space-time point where now space can be arbitrary number of dimensions, we will write the two sets which represents the future and past light rays emanating from our point. We see that they form two cones, intersection only at the vertex, in space-time. We will also denote the sets to be respectively the future of and past of .)

Another thing to read off from the picture, and from Equations 18 and 19, is that the solution will not “settle down to a constant value” until the support set of the inhomogeneity is completely contained in . So in particular, if the support set for has infinite spatial (or temporal) extent, then the solution will see “tail behaviour”. This supports the interpretation, using Duhamel’s Principle, of the right-hand side of the wave equation as a “source term”.

A quick detour to Penrose diagram

To better illustrate the global behaviour of wave equations, we will use the so-called (Carter-)Penrose diagrams for space-time. To study late time behaviour, we will need to think about points infinitely far into the future. And so we want a convenient way to represent the space-time and its infinite extent on a finite piece of paper. This requires some creative rescaling of space-time coordinates. Now, there are many ways to map the infinite to the finte. For example, consider the tangent function. It is a one-to-one mapping between the open interval and the entire real line. The Penrose diagram is a particular type of mapping. Now, if you look at any mapping from , representing the 1+1 dimensional space-time, to a finite region, it would be necessary that some of the straight-lines be mapped to lines that are no longer straight. (If all straight lines are preserved, the mapping must be affine by the fundamental theorem of projective geometry.) So the question is what, if any, curves we want to keep “straight”. As we have already seen, what is most important in the study of wave equations is the null lines ; they, being characteristics of the partial differential equation, encode most of the phenomenon that we want to study. The Penrose diagram is a way of representing the 1+1 dimensional Minkowski space-time as a set with finite extent, such that the null lines (which are also the level sets of the coordinate functions) are preserved.

Figure 3

Figure 3 shows some examples of Penrose diagrams. (a) is the Penrose diagram for the 1+1 dimensional Minkowski space, the boundaries of the diamond are precisely where the coordinate functions attain values; we include the diagonal lines to indicate the level sets of . (b) is the Penrose diagram for the Minkowski half space; this also corresponds to the Penrose diagram for the 1+3 dimensional Minkowski space after we quotient out by the spherical symmetry. Notice that the central axis is drawn slightly curved to illustrate the fact that non-null straight lines don’t necessarily map to straight lines in the compactified diagram. In (a) and (b) we see that the open circles and the dashed lines represent “idealized boundaries”, points that are not in fact in the space-time; in (b) the solid line for the central axis indicates the fact that boundary is in fact a physical region of the space-time. (c) is a representation of Figure 2 in compactified form; observe that the bottom boundary of the blue triangle is drawn slightly curved also. (d) shows the “domain of influence” of a source. In the picture the cyan ball indicates the space-time support set of the inhomogeneity. Under the assumption of either Dirichlet boundary condition (if we consider this as a 1+1 problem on the Minkowski half space) or symmetric reduction from a 1+3 dimensional problem, the pink region is the region in space-time where a priori, the contribution from the inhomogeneity to the solution of the wave equation can be non-zero. The larger the space-time extent of the inhomogeneity, the larger the domain of influence.

Nonlinear tails: an example

Be start by giving the motivation for the construction. Consider the following Penrose diagram, in which we study the future evolution of some wave equation in 1+1 dimensional space with Dirichlet boundary (which could be the symmetry reduction of a spherically symmetric wave equation in 1+3 dimensional Minkowski space).

Figure 4

The initial data is prescribed to be only non-vanishing on the blue line. So if we were to solve the purely linear homogeneous problem, using what we discussed in the previous post in the series, we see that the initial data splits into an incoming part, which approaches the central axis, bounces off, and then radiates to infinity; and an outgoing part which immediately radiates outward. And as we described before, following any future directed time like curve (which can be represented by a curve converging to the top open circle, the “future timelike infinity”) we will eventually escape the region influenced by this data.

Equivalently, we can consider the value of the solution at a point in the space time (here the red dot). Using the representation given by Duhamel’s principle, we see that for the inhomogeneous wave equation with possible source term, the solution can only be non-vanishing at the red dot if the initial data and any inhomogeneities have space-time support that intersects the crosshatched region. So for our linear homogeneous problem, where the initial data (blue curve) does not intersect the crosshatched set determined by strong Huygens’ principle, at the red dot the solution would vanish.

Now suppose that instead of an outside source, we impose a nonlinear self-interaction on the right hand side. That is, the source term is assumed to be generated from non-linear effects of the wave itself. To be slightly more precise, assume our equation takes the form

Equation 20

where is now only a function of the value of the displacement , with (these conditions ensure that the interaction is non-linear, and that waves won’t be spontaneously generated from 0 initial data; in other words, 0 is still a solution to the equation).

Now, because the interaction is nonlinear, if is small, we can expect, by Taylor expanding , that the source term contribution would be , which is even smaller. Thus intuitively we expect to be able to solve the equation using successive approximation, namely: we start with the given initial data , we solve the homogeneous wave equation for it, and call the solution . Then we run through the following iterative procedure:

1. We are fed the solution from the previous iterate, .
2. We solve the inhomogeneous wave equation with prescribed source: subject to the initial data . Notice that on the right hand side, the source is given by , with on the left hand side, we are solving for . So we can directly obtain an integral expression using Duhamel’s principle.
3. Feed into this algorithm and repeat.

If this approximation scheme converges, then we will have that , the limiting function, will solve the nonlinear equation. (See Appendix for a bit more details.)

Let us consider for now the first iterate, which is enough to give a heuristic demonstration of why nonlinear tails are generic, and that the complete vanishing of waves within finite time that is possible in the linear system is very unstable.

Going back to Figure 4, suppose the initial data is precisely supported on the blue line. Then the 0th iterate, , just solves the linear homogeneous wave equation, and so its support set is represented by the cyan region, and the red dot lies outside of it. But now let us consider the first iterate. It solves the wave equation with source term ! So in particular, while it is true that the crosshatched region does not intersection the initial data (so there is no contribution to at the red dot coming from the prescribed initial value ), the region does intersect the cyan set, on which our inhomogeneity does not identically vanish. And if the homogeneity is assumed to take only non-negative values (for example ), then it is necessary that is nonvanishing at the red dot. Since the position of the red dot is more or less arbitrary, we see that at the level of the first iterate, we already expect some nonlinear tails.

Appendix: successive approximation and local existence of solutions for nonlinear problems

Observe that using Duhamel’s principle we have a very precise control of the rate of convergence. Consider the difference . It solves the wave equation

with 0 initial condition. The right hand side we again use Taylor expansion (assuming that is small), to replace by

But since is also small, we can approximate by . So by the assumption that , we see that the source term is small. In particular, applying Duhamel’s principle again, using that the integral used to factor in the inhomogeneity is taken over a finite space-time region, we have that

where the constant depends only on the maximum time . Using that , we see that is in fact controlled by a geometric series, and the series converges if is bounded by an such that is less than, say, 1/100.

Equation 1

where is a real valued function defined on (1+n)-dimensional Minkowski space that describes our solution, and represents a “source” term. When vanishes identically, we say that we are looking at the linear, homogeneous wave equation. When is itself a function of and its first derivatives, we say that the equation is a semilinear wave equation.

We first start with the homogeneous, linear case.

Homogeneous wave equation in one spatial dimension

One interesting aspect of the wave equation is that it only possesses the second, multidimensional, dispersive mechanism as described in my previous post. In physical parlance, the “phase velocity” and the “group velocity” of the wave equation are the same. And therefore, a solution of the wave equation, quite unlike a solution of the Schroedinger equation, will not exhibit decay when there is only one spatial dimension (mathematically this is one significant difference between relativistic and quantum mechanics). In this section we make a computation to demonstrate this, a fact that would also be useful later on when we look at higher (in particular, three) dimensions.

Use for the variable representing spatial position. The wave equation can be written as

Now we perform a change of variables: let and be the canonical null variables. The change of variable formula replaces

Equation 2

and we get that in the coordinate system,

Equation 3

Equation 3 implies that the derivatives of solve transport/ordinary differential equations. More precisely, it says that is independent of , and using the commutation of second derivatives, is independent of . Hence on each instant in time , the maximum value of and the maximum value of are independent of . Which is the first indication that we do not have amplitude decay.

To be more explicit, we can derive an explicit representation of the solution to the initial value problem. Assume that at time , we have that . We can decompose into forward and backward traveling waves

Equation 4
with

The second conditions guarantee that the corresponding solution decomposition (by linearity of the Equation 1) satisfies . So we can write

Equation 5

Solving for the algebraic conditions in Equation 4, we have which implies

Equation 6

where the initial point of integration is fixed arbitrarily to 0. Combining this with Equation 5 we get that

Equation 7

From this we can derive the following fact about non-decay of solutions to the wave equation. For simplicity we will assume our initial data are smooth functions, and are compactly supported in the interval ; similar statements with the usual modifications can be had if we assume are Schwartz distributions with sufficiently fast decay.

First we consider the statement with fixed . By assumption of compact support, for , . And in this case the solution becomes

In fact, a modification of the same argument shows that if is a -curve such that there exists some for which everywhere (so that the space-time curve is uniformly time-like), then there is a maximum time (depending on and ) such that

for any .

So for generic initial data where the total integral of is non-vanishing, along every uniformly time-like trajectory, eventually the wave will settled down to a non-zero constant displacement. Furthermore, while for the case we have vanishing integral that at every point on the line the wave will eventually settle down to 0, we have that the supremum over a time slice is never vanishing. To see this, notice that in the definition for in Equation 6, we can instead set the lower bound of integration to . Then if has compact support and vanishing total integral, we can write , where is a smooth function with compact support also, and . Now if , we also have , and hence at least one of is non-vanishing. Let be a point such that , then we have that

if

so at each time there is lower bound (independent of time) to the maximum amplitude achieved by the wave at that time. In Figure 1 below, we illustrate the above. The initial data is supported on the axis in the region highlighted in green. The pink region is where the wave settles down to the constant value given by the total integral of . The solutions in the cyan regions are given by translations of , with the right hand region depending only on the forward wave , and the left hand region on the backward wave. If is non-zero, then in the right hand cyan region and are non-zero, and independent of ; similarly in the left hand cyan region for the backward wave. The blue curve represents , a time-like trajectory that eventually enters the “interior” pink region.

Figure 1

In this picture, the lack of dispersion can be seen in the fact that the cross sectional area at each fixed time of the cyan-colored “wave zone” (remember that outside this wave zone the solution is constant) is bounded above by twice the area of the support of the initial data .

Dirichlet boundary in one spatial dimension

Before moving on to higher dimensions, we will have to first consider one dimensional waves on a one-sided domain with Dirichlet boundary conditions. To be more precise, we now take to be a function of time and space, where the spatial parameter runs from . When an equation is prescribed on a domain with boundary, we need to also specify how the wave must behave at the boundary. For purposes of application to higher dimensions, we will consider the case of Dirichlet boundary condition, which states that for all time. Physically this represents a wave bouncing off a hard boundary (like, say, sound waves bouncing off a wall).

One can try to solve the equation directly and impose the Dirichlet boundary condition. But it may be more instructive to use this opportunity to illustrate the principle of using symmetry properties to our aid. Notice that the homogeneous linear wave equation, by virtue of its linearity, has the principle of superposition. So one way of dealing with the Dirichlet boundary condition is to extend the spatial domain to negative values of and mirror the solution about the origin. To be more precise:

Let us take to be initial data, compactly supported in , to the initial boundary value problem with Dirichlet boundary conditions at . Instead of dealing with the boundary condition, we can instead consider the initial value problem on the whole real line (as discussed with previous section) for the wave equation, with initial data given by

Equation 8
if
if

One can easily check that the solution to this modified problem vanishes identically at . And so if we restrict to the positive half line, it is in fact a solution to the Dirichlet initial boundary value problem that we started with. (To show that this is the only possible solution requires a bit of care, but can be done; we’ll omit it here.)

(Interesting to note that if instead of Equation 8 we take when , we would solve the corresponding Neumann initial boundary value problem, where the boundary condition is . This physically corresponds to the case where the medium is completely flexible at the boundary.)

One particularly nice aspect of this way of looking at the Dirichlet problem is that we can apply the previous results on wave equations on the whole real line. In particular, by construction the initial data must have vanishing total integral. So for any fixed , we can find some finite time such that we will have for any .

The homogeneous case with 3 spatial dimensions

The more physically interesting case is when we have 3 spatial dimensions, which agrees with our intuitive understanding of the world. For convenience of the discussion, we will restrict ourselves to considering spherically symmetric functions; this is in general sufficient to give the intuition behind what I want to discuss. In the particular case of Minkowski space, one can also transfer, using the method of spherical means, the discussion here about spherically symmetric to statements about the fundamental solution, and hence also statements about general solutions of the linear homogeneous wave equation. To how see this pans out, a good reference is Sigmund Selberg’s Johns Hopkins lecture notes on wave equations.

First let us consider the reduction of the linear wave equation. In three dimensions, we can write the Laplace operator in spherical coordinates , where is the radial coordinate and represents some 2-dimensional coordinate system on the unit sphere (its exact form is unimportant, since we will be working in spherical symmetry).

where we use to denote the induced Laplace operator on the unit sphere. In spherical symmetry, is independent of the spherical coordinates, so any derivatives tangential to the sphere vanish, in particular . We can do a further simplification, however. Consider the function . We have that

while

so we have that, for a spherically symmetric function ,

On the other hand, we have that, since ,

and so we arrive at the reduced wave equation

Equation 9
for where solves Equation 1

Note that the choice of weight that one uses to define is not coincidental: that we use to the first power is a reflection of the typical behaviour expected of waves in three spatial dimensions. Imagine a radio wave spreading out from the radio tower. The energy density of the wave is proportional to square of the amplitude (which is why for computation of electrical power of alternating currents we use RMS). But the wave propagates at roughly constant speed outwards radially. So after the wave has travelled a distance , the the spherical shell representing the wave front has area proportional to , while the total energy is conserved. So this means that the energy density (and hence power density) drops as inverse squared of the distance from the tower. This means that times the amplitude should roughly be constant. In spatial dimension , the same argument gives that the expected weight should be , but the computation in general is not as simple and pretty as the three-dimensional case.

The important thing about Equation 9 is that solves a 1 dimensional wave equation on the half plane and . What are the boundary conditions? If the solution remains regular, it must remain bounded at the origin. Since and at the origin, we have that the boundary condition must be Dirichlet. So in particular, we can apply the result of the previous section to the case of the homogeneous equation, .

Theorem 1 Strong Huygens’ Principle
Let solve the homogeneous wave equation on (1+3) dimensional Minkowski space. Suppose we have that the initial data for is supported within the ball of radius around the origin. Then must be identically vanishing in the following two regions:

• The exterior region .
• The Interior region .

In other words, the solution is only supported on the wave zone .

The exterior region corresponds roughly to the pink region in Figure 1, while the wave zone would be roughly the cyan regions. Note that the sketch of proof outlined above proceeds via spherical symmetry, and so we have only demonstrated that the theorem is true in the spherical symmetric case; as mentioned before by using the methods of spherical means the theorem can be shown to be true for general solutions.

What we have studied have two important consequences: using the results from the previous section, we see that on each constant time slice, for initial data that is compactly supported, the maximal amplitude of the wave is constant (after some initial time to let the “incoming” and “outgoing” components of the wave separate). This means that the function of time

Equation 11

where we use the fact that is supported only in the wave zone, where , so . So that “over all” the wave decays in amplitude like . On the other hand, we also have the consequence of Theorem 1, which says that

Equation 12
such that whenever

so in particular at any fixed point in space, after waiting a sufficiently long period of time, you will eventually feel nothing but peace and quiet.

As it turns out, Equation 11 is somewhat stable: for nice inhomogeneity (which allows certain types of non-linearities, if we assume the initial data is “small”) we can recover the decay rate. But Equation 12, however, is extremely unstable. Recall from the previous discussion the phenomenon of tails. Equation 12 basically says that for the linear, homogeneous wave equation on Minkowski space with compactly supported initial data, there can be no tails. As our intuition suggests from the previous post, if we add “obstacles” or “sources” (which can be captured in the function ) there will be backscattering from the out going waves, and this will lead to some waves persisting after long periods of time. Furthermore, Equation 12 also strongly depends on having compactly supported initial data. One way of thinking about Equation 12 is that the particularly nice features of Minkowski space allows one to convert “decay in space at a fixed moment in time” to “decay in time at a fixed point in space”. By assuming lots and lots of decay in space, we get also lots and lots of decay in time. But any reduced decay in space will automatically translate to also a reduced decay in time.

(If there is no decay in space, there can also be no decay in time: observe that the constant function is a solution to the homogeneous wave equation.)

And I hate to admit defeat to technology.

This first post will not be too heavily mathematical, indeed, the only realy foray into mathematics will be in the appendix; the next ones, however, requires some basic familiarity with partial differential equations and pseudo-Riemannian geometry.

Dispersion and decay

The wave equation, as its name suggests, describes the propagation of waves. In the context of the discussion here, the precise nature of the wave is unimportant: we don’t care so much whether the wave is longitudinal or transversal, we don’t need to know what the medium is that facilitates the wave propagation (or whether such a medium is necessary at all), nor do we wonder about what caused the wave in the first place. These kinds of differences manifest themselves mathematically as internal nonlinear structures, and in the limit when the amplitude of the wave is vanishingly small, these nonlinear effects can be “safely disgarded” (not strictly true, see the next section). But as a concrete example to keep in the back of the mind, we could imagine ripples on the surface of a pond, or radio (electromagnetic) waves.

The important thing for the present discussion is that waves disperse. What do we mean by dispersion? Imagine dropping a pebble into an absolutely still pond. The ripples form concentric rings that spread out. That is dispersion. Visualise a radio beacon broadcasting a signal. The signal starts with high power at the source but spreads out spherically, attenuating with distance. That is dispersion. For contrast, imagine an ideal laser beam: the beam is perfectly collimated and the cross section area is the same 1 meter, 1 kilometer, or 1 lightyear away from the source. That is the lack of dispersion.

Very roughly speaking, dispersion is the tendency for the wave to occupy more and more area as it propagates. In the case of ripples on the pond, the measure of “area” is the circumference of the ripple. When the pebble is first dropped in, a high amplitude disturbance is created at the spot where the pebble hit the water. As time passes by, you get concentric ripples flowing outwards whose amplitude decreases as the radius of the ripples grow.

This decay in amplitude, as it turns out, is characteristic of dispersive systems. The reason underpinning this connecting between dispersion and decay is the fundamental principle of “conservation of energy” (see also the appendix to this post). The wave carries with it some finite amount of total energy, which, being conserved, can be computed at the initial time. As the support of the wave (where the disturbance is non-zero) spreads out, the energy has to be distributed among larger and larger area. This means that the energy density, being roughly the average of the total energy over the area in which the wave is non-zero, would decay over time. Converting from energy density back to amplitude, we see that a wave with conserved total energy that exhibits dispersive phenomenon must decay in time. (Note that this decay is “local” and is different from the decay associated from dissipation associated to friction, viscosity, or damping. In particular, to “globalize” this decay far into the future, we need the area covered by the wave to keep increasing in time. If the spatial domain has limited extent, then the maximum area supporting the wave also is limited. Imagine again the pebble in the pond. If the water were not viscuous at all, and there’s no mechanical loss to friction or heat in the fluid, then the ripples will eventually reach the shore and bounce back. After a long time the ripples and their reflections will cover the surface of the entire pond and then it won’t decay further in amplitude. Another example is the phenomenon of standing waves. Absent friction effects, standing waves have more-or-less constant amplitude, reflecting the constraint of bounded spatial domain.)

So what causes dispersion? In general it is best described as the tendency of a wave to decompose into components traveling with different velocities. There are generally two ways for this to happen. The first is a one-dimensional effect, where the wave is separated into components traveling at different speeds. It is like a footrace: all racers start at the same place (the starting line) and run in the same direction; but at the end of the race the racers are spreadout, with some reaching the finishline before others. Oftentimes this decomposition is based on the frequency of the wave. A striking illustration of this is the Newtonian experiment splitting light into the rainbow of colors. In the glass prism, different frequencies of light travel at different speeds. Coupled with Snell’s law, and the relation between refractive index and light speed, this forces white light, which is a blend of electromagnetic waves of different frequencies, to split up into separate components travelling at different speeds and angles.

The second dispersive effect is a multi-dimensional effect, where the wave is separated into components traveling in different directions. This is more akin to what we imagine on the surface of a pond, or of radio waves: the spreading out is because we have a bunch of little packets of energy going all in different directions. For systems satisfying the linear wave equation, this second dispersive effect is what causes decay. And thus a solution to the wave equation in one dimension (a wave traveling along a string) does not decay in amplitude. A system that also makes use of the first dispersive effect is the linear Schroedinger equation of quantum mechanics, for which the spreading out of the wave function, even in one spatial dimension, can be seens as a manifestation of the uncertainty principle.

Scattering and tails

Now imagine instead of a small pond, an absolutely still ocean of infinite extent. We drop a rock into it, and a ripple travel outward. It gets further and further away, and eventually disappears from sight: all the energy has been carried away by the wave and there is no disturbance left where the rock fell in. Now, however, imagine that our ocean is dotted with stone pillars that rises out of the water. Again we drop in a rock, the ripples flow outwards and suddenly it bumps into a pillar! Instead of continuing, the wave bounces off the pillar and flows outward from the pillar as if it were the center of the disturbance. This reflection of the wave off of obstacles is the first example of scattering.

In general, scattering does not have to be due to “hard” obstacles like the pillars in the ocean. Reflection can also happen across transition boundaries between two media. A good example is exhibited by one-way mirrors. At the boundary between glass and air (or metal coating and air), an incoming electromagnetic wave is partially reflected and partially transmitted. When one side of the glass is much brighter than the other, the intensity of the reflected light, when viewed from the bright side, albeit weak, is still stronger than the intensity of the light transmitted from the dark side. Whereas from the dark side the light transmitted through the glass from the bright side is much brighter than the light reflected from within the dark room. This type of scattering doesn’t have to be across a discontinuous boundary either. In general one will expect some sort of back scattering of waves when the medium itself is not homogeneous.

As already seen in the case with the stone pillars in the ocean, the phenomenon of back scattering will cause waves which “ought of have left the area” to remain. In the extreme case, where the entire medium is encased in reflective material (a pond of finite extent, a radio wave guide), we will have non-decay of the wave, in spite of the dispersive effects. In the intermediate regime, the wave will still (mostly) escape when given long enough time, but this remnant from back scattering will cause the associated decay to be slower than one may have predicted using a purely local description coming from studying a homogeneous medium. This is what we call a tail.

Appendix: two “derivations” of the wave equation

Let be the function that describes the wave amplitude. One can think of it as, for example, the vertical displacement, as a function of position and time , of a horizontally held elastic membrane (say a drum head).

The Newtonian picture. In this picture, the basic idea is the restorative force. Recall Newton’s second law, which states that the acceleration experienced by a mass is proportional to the force acting on it. Or in notation, , where is the function representing the force per unit mass experienced at position and time . For wave motion, we postulate that the restorative force is proportional to “how much more displaced the particle at position is, when compared against its immediate neighbors”. This latter quantity can be mathematically described as the difference between , and , the average of over a small ball . Taking the multivariable Taylor expansion of where is the Hessian operator, we compute its average on a small ball. Notice that for all odd terms (meaning a function such that there exists a hyperplane with being the negative of its reflection across the plane), the average must be zero. So up to second order, the only terms that contribute to the average is and , where denotes the Laplacian operator. So taking the difference between and , we have that the restorative force is proportional to , where I put the subscript on the Laplacian operator to remind ourselves that the Laplacian is with respect to the spatial variables. Lastly, we need to know the constant of proportionality. In general this is determined by the media in question, as well as by the units of measurement. However, we can determine the sign of proportionality constant. If a point is more (positively) displaced than all its neighbors, it is a local maximum, and so its second derivative, and hence its Laplacian, must be negative. Since the force is restorative, it will need to point in the negative direction also, and so we have that tha proportionality constant has a positive sign. That is, the prototype of the wave equation is

The Hamiltonian picture. Here, we do the same derivation, but using as our guide, instead of Newton’s second law of motion, the law of conservation of energy. For a closed system, we must have that the sum of kinetic and potential energies is constant over time. But for an open system, the total energy can change by taking in or expelling a flux through the boundary of the system. Infinitesimally, this is described by the equation . Here is the energy density, is the energy flux vector, and $\nabla\cdot$ the vector divergence, and the Hamitonian picture reduces to the specification of what are the constitutive equations (or equations of state) which tell us how and can be computed from the displacement .

To compute the energy density, we observe that the kinetic energy will take the form as usual, coming from the velocity of each point of the medium. The potential energy we will assume is given by Hooke’s Law, which in infinitesimal form implies . (We can again use the Taylor expansion to arrive at this: for a spring, Hooke’s law gives potential energy where is the displacement from equilibrium. We can now compute the average over a small ball of to give the potential energy for the wave.) is the mass density and is the analogue of the spring constant. For the energy flux, it is well known that the energy transfered to/from a moving particle is given by the product of the force acting on the particle and its velocity. Using Hooke’s law again, we then have . Plugging everything into the conservation law, we end up with (after some cancelling of terms on the left and right hand sides),