### Decay of waves I: Introduction

In the next week or so, I will compose a series of posts on the heuristics for the decay of the solutions of the wave equation on curved (and flat) backgrounds. (I have my fingers crossed that this does not end up aborted like my series of posts on compactness.) In this first post I will give some physical intuition of why waves decay. In the next post I will write about the case of linear and nonlinear waves on flat space-time, which will be used to motivate the construction, in post number three, of an example space-time which gives an upper bound on the best decay that can be generally expected for linear waves on non-flat backgrounds. This last argument, due to Mihalis Dafermos, shows that why the heuristics known as Price’s Law is as good as one can reasonably hope for in the linear case. (In the nonlinear case, things immediately get much much worse as we will see already in the next post.)

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 $\Phi$ be the function that describes the wave amplitude. One can think of it as, for example, the vertical displacement, as a function of position $x$ and time $t$, 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, $\ddot{Phi}(t,x) = f(t,x)$, where $f(t,x)$ is the function representing the force per unit mass experienced at position $x$ and time $t$. For wave motion, we postulate that the restorative force is proportional to “how much more displaced the particle at position $x$ is, when compared against its immediate neighbors”. This latter quantity can be mathematically described as the difference between $\Phi(x)$, and $\bar{\Phi}(x)$, the average of $\Phi(x+y)$ over a small ball $|y| \leq \epsilon$. Taking the multivariable Taylor expansion of $\Phi(x+y) = \Phi(x) + y\cdot \nabla \Phi(x) + \frac{1}{2}y\cdot \left(\nabla^2 \Phi(x)\right)\cdot y + \ldots$ where $\nabla^2$ is the Hessian operator, we compute its average on a small ball. Notice that for all odd terms (meaning a function $g$ such that there exists a hyperplane with $g(y)$ 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 $\Phi(x)$ and $\frac{1}{2}|y|^2 \triangle \Phi(x)$, where $\triangle$ denotes the Laplacian operator. So taking the difference between $\Phi(x)$ and $\bar{\Phi}(x)$, we have that the restorative force is proportional to $\triangle_x \Phi(t,x)$, 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 $x$ 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

$\ddot{\Phi}(t,x) = k^2 \triangle_x \Phi(t,x)$

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 $\dot{e}(t,x) = \nabla\cdot j(t,x)$. Here $e$ is the energy density, $j$ 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 $e$ and $j$ can be computed from the displacement $\Phi$.

To compute the energy density, we observe that the kinetic energy will take the form $\frac{1}{2}\mu \dot{\Phi}^2$ 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 $\frac{1}{2}\kappa |\nabla\Phi|^2$. (We can again use the Taylor expansion to arrive at this: for a spring, Hooke’s law gives potential energy $\frac{1}{2}\kappa d^2$ where $d$ is the displacement from equilibrium. We can now compute the average over a small ball of $\frac{1}{2}\kappa (\Phi(x+y) - \Phi(x))^2$ to give the potential energy for the wave.) $\mu$ is the mass density and $\kappa$ 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 $j = \kappa \dot{\Phi}\nabla\Phi$. Plugging everything into the conservation law, we end up with (after some cancelling of terms on the left and right hand sides),

$\mu \dot{\Phi} \ddot{\Phi} = \kappa \dot{\Phi} \triangle_x\Phi$