### Decay of waves IIa: Minkowski background, homogeneous case

Now let us get into the mathematics. The wave equations that we will consider take the form

Equation 1
$-\partial_t^2 \phi + \triangle \phi = F$

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

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 $x\in\mathbb{R}$ for the variable representing spatial position. The wave equation can be written as

$-\partial_t^2 \phi + \partial_x^2\phi = 0$

Now we perform a change of variables: let $u = \frac{1}{2}(t-x)$ and $v = \frac{1}{2}(t+x)$ be the canonical null variables. The change of variable formula replaces

Equation 2
$\displaystyle \partial_t \to \frac{\partial u}{\partial t} \partial_u + \frac{\partial v}{\partial t} \partial v = \frac{1}{2}\partial_u + \frac{1}{2}\partial_v$
$\displaystyle \partial_x \to \frac{\partial u}{\partial x} \partial_u + \frac{\partial v}{\partial x} \partial v = -\frac{1}{2}\partial_u + \frac{1}{2}\partial_v$

and we get that in the $(u,v)$ coordinate system,

Equation 3
$-\partial_u \partial_v \phi = 0$

Equation 3 implies that the derivatives of $\phi$ solve transport/ordinary differential equations. More precisely, it says that $\partial_v\phi$ is independent of $u$, and using the commutation of second derivatives, $\partial_u\phi$ is independent of $v$. Hence on each instant in time $u+v = \tau$, the maximum value of $|\partial_v\phi|$ and the maximum value of $|\partial_u\phi|$ are independent of $\tau$. 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 $t = 0 = u+v$, we have that $\phi(0,x) = f(x), \partial_t\phi(0,x) = g(x)$. We can decompose $(f,g)$ into forward and backward traveling waves

Equation 4
$(f,g) = (f_+,g_+) + (f_-, g_-)$ with $\partial_xf_+ + g_+ = 0 = -\partial_x f_- + g_-$

The second conditions guarantee that the corresponding solution decomposition (by linearity of the Equation 1) $\phi = \phi_+ + \phi_-$ satisfies $\partial_v \phi_+ = 0 = \partial_u \phi_-$. So we can write

Equation 5
$\phi(t,x) = \phi_+(t,x) + \phi_-(t,x) = \phi_+(0,x-t) + \phi_-(0,x+t) = f_+(x-t) + f_-(x+t)$

Solving for the algebraic conditions in Equation 4, we have $\partial_x f_{\pm} \mp g_{\pm} = \partial_x f \mp g$ which implies

Equation 6
$\displaystyle f_{\pm}(x) = \frac{1}{2}\left(f(x) \mp \int_{0}^x g(y) dy \right)$

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

Equation 7
$\displaystyle \phi(t,x) = \frac{1}{2}\left( f(x-t) + f(x+t) + \int_{x-t}^{x+t} g(y) dy \right)$

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 $(f,g)$ are smooth functions, and are compactly supported in the interval $[-1,1]$; similar statements with the usual modifications can be had if we assume $(f,g)$ are Schwartz distributions with sufficiently fast decay.

First we consider the statement with fixed $x = x_0$. By assumption of compact support, for $t > |x_0| + 1$, $f(x_0 - t) = f(x_0 + t) = 0$. And in this case the solution becomes

$\displaystyle \phi(t > |x_0| + 1, x_0) = \frac{1}{2}\int_{-1}^{1} g(y) dy$

In fact, a modification of the same argument shows that if $\gamma:\mathbb{R}\to\mathbb{R}$ is a $C^1$-curve such that there exists some $\epsilon > 0$ for which $|\dot{\gamma}| \leq 1 - \epsilon$ everywhere (so that the space-time curve $(t,\gamma(t))$ is uniformly time-like), then there is a maximum time $\tau_M$ (depending on $\gamma(0)$ and $\epsilon$) such that

$\displaystyle \phi(t, \gamma(t)) = \frac{1}{2}\int_{-1}^{1} g(y) dy\qquad$ for any $|t| > \tau_M$.

So for generic initial data where the total integral of $g$ 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 $g$ 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 $f_{\pm}$ in Equation 6, we can instead set the lower bound of integration to $-\infty$. Then if $g$ has compact support and vanishing total integral, we can write $g = G'$, where $G$ is a smooth function with compact support also, and $f_{\pm} = f \mp G$. Now if $(f,g) \not\equiv 0$, we also have $(f,G) \not\equiv 0$, and hence at least one of $f_{\pm}$ is non-vanishing. Let $x_0$ be a point such that $\max(|f_+(x_0)|,|f_-(x_0)|) > 0$, then we have that

$\sup_{x} \left|\phi(t,x)\right| \geq \max( |\phi(t, x_0 + t)|, |\phi(t,x_0 - t)|)$ $= \max( |f_+(x_0)|,|f_-(x_0)|) > 0\qquad$ if $|t| > 2$

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 $x$ 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 $g$. The solutions in the cyan regions are given by translations of $f_{\pm}$, with the right hand region depending only on the forward wave $f_+$, and the left hand region on the backward wave. If $f_+$ is non-zero, then in the right hand cyan region $\phi$ and $\partial_u\phi$ are non-zero, and independent of $v$; similarly in the left hand cyan region for the backward wave. The blue curve represents $(t,\gamma(t))$, 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 $\phi$ is constant) is bounded above by twice the area of the support of the initial data $(f,g)$.

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 $\phi$ to be a function of time and space, where the spatial parameter runs from $[0,\infty)$. 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 $\phi(t,0) = 0$ 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 $x$ and mirror the solution about the origin. To be more precise:

Let us take $(f,g)$ to be initial data, compactly supported in $(0,\infty)$, to the initial boundary value problem with Dirichlet boundary conditions at $x = 0$. 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 $(\tilde{f},\tilde{g})$ given by

Equation 8
$(\tilde{f},\tilde{g})(x) = (f,g)(x)\qquad$ if $x > 0$
$(\tilde{f},\tilde{g})(x) = (-f,-g)(-x)\qquad$ if $x < 0$

One can easily check that the solution $\tilde{\phi}$ to this modified problem vanishes identically at $x = 0$. And so if we restrict $\tilde{\phi}$ 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 $(\tilde{f},\tilde{g})(x) = (f,g)(-x)$ when $x < 0$, we would solve the corresponding Neumann initial boundary value problem, where the boundary condition is $\partial_x\phi(t,0) = 0$. 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 $\tilde{g}$ must have vanishing total integral. So for any fixed $x_0 \geq 0$, we can find some finite time $\tau$ such that we will have $\phi(t,x_0) = \tilde{\phi}(t,x_0) = 0$ for any $|t| > \tau$.

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 $(r,\omega)$, where $r\in \mathbb{R}_+$ is the radial coordinate and $\omega \in \mathbb{S}^2$ represents some 2-dimensional coordinate system on the unit sphere (its exact form is unimportant, since we will be working in spherical symmetry).

$\displaystyle \triangle \phi = \frac{1}{r^2} \frac{\partial}{\partial r} r^2 \frac{\partial}{\partial r}\phi + \frac{1}{r^2} \Omega^2\phi$

where we use $\Omega^2$ to denote the induced Laplace operator on the unit sphere. In spherical symmetry, $\phi(r,\omega) = \phi(r)$ is independent of the spherical coordinates, so any derivatives tangential to the sphere vanish, in particular $\Omega^2\phi = 0$. We can do a further simplification, however. Consider the function $\psi = r\phi$. We have that

$\partial_r\partial_r \psi = \partial_r (r \partial_r\phi + \phi) = r \partial_r\partial_r\phi + 2 \partial_r \phi$

while

$\partial_r r^2 \partial_r \phi = 2 r \partial_r \phi + r^2 \partial_r\partial_r \phi$

so we have that, for a spherically symmetric function $\phi$,

$\displaystyle \triangle \phi = \frac{1}{r} \partial_r^2 \psi$

On the other hand, we have that, since $\partial_t r = 0$,

$\displaystyle \frac{1}{r}\partial_t^2 \psi = \partial_t^2 \phi$

and so we arrive at the reduced wave equation

Equation 9
$-\partial_t^2 \psi + \partial_r^2\psi = r F \qquad$ for $\psi = r\phi$ where $\phi$ solves Equation 1

Note that the choice of weight that one uses to define $\psi$ is not coincidental: that we use $r$ 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 $r$, the the spherical shell representing the wave front has area proportional to $r^2$, 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 $r$ times the amplitude should roughly be constant. In spatial dimension $N$, the same argument gives that the expected weight should be $r^{\frac{N-1}{2}}$, but the computation in general is not as simple and pretty as the three-dimensional case.

The important thing about Equation 9 is that $\psi$ solves a 1 dimensional wave equation on the half plane $t\in (-\infty,\infty)$ and $r \in (0,\infty)$. What are the boundary conditions? If the solution $\phi$ remains regular, it must remain bounded at the origin. Since $\psi = r\phi$ and $r = 0$ 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, $F = 0$.

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

• The exterior region $|x| > R + |t|$.
• The Interior region $|t| > R + |x|$.

In other words, the solution $\phi$ is only supported on the wave zone $\left| |t| - |x| \right| \leq R$.

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 $\psi$ is constant (after some initial time $t_0$ to let the “incoming” and “outgoing” components of the wave separate). This means that the function of time

Equation 11
$\displaystyle \sup_{x} |\psi(t,x)| \sim C$
$\displaystyle \implies \sup_{x} |\phi(t,x)| \leq \frac{C}{|t|}$

where we use the fact that $\phi$ is supported only in the wave zone, where $|t| \sim |x|$, so $|\psi| = |r\phi| \sim |t\phi|$. So that “over all” the wave decays in amplitude like $t^{-1}$. On the other hand, we also have the consequence of Theorem 1, which says that

Equation 12
$\forall x, \exists \tau = \tau(x)$ such that $\phi(t,x) = 0$ whenever $|t| > \tau(x)$

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 $F$ (which allows certain types of non-linearities, if we assume the initial data is “small”) we can recover the $t^{-1}$ 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 $F$) 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 $\phi(t,x) = C$ is a solution to the homogeneous wave equation.)