4.10 Reading the solutions — what the wave equation does

The previous nine lessons derived

four separate ways. Deriving it tells us why it holds. This lesson tells us how to read it: what its solutions can and cannot do, and which feature of the equation is responsible for which physical fact. The same equation — with $p'$ standing for a string’s displacement, an electromagnetic field, a seismic disturbance, or a quantum amplitude — governs all of them, so what follows is the operating manual for waves in general, written in the language of sound.

It is a guided tour. Each idea below gets one sentence of statement and one live demonstration; each also points to the later lesson that develops it in full. Treat the page as the map for chapters 5 through 10, and the summary as the territory drawn small.

1. Curvature causes acceleration

Read the equation as a sentence. The left side is an acceleration; the right side is a curvature — $\nabla^2 p'$ measures how far $p'$ at a point sits below the average of its neighbours. The equation makes them proportional: a point accelerates because its neighbours are out of line with it. A pressure valley is pulled up, a crest is pushed down. That is the entire local mechanism; everything else is consequence.

The three symbols divide the labour cleanly: $c$ holds the medium (through $c^2 = (\partial p/\partial \rho)_s$, lesson 4.9), $\nabla^2$ holds the space, and the boundary conditions — still unwritten — hold the object. Change the gas and only $c$ moves; change the container and only the boundaries move. One equation, a violin string and a concert hall.

2. Disturbances travel at a finite speed

Everything in this section follows from one fact: a disturbance spreads through the medium at a fixed speed $c$, and no faster. d’Alembert’s solution $p'(x,t) = f(x - ct) + g(x + ct)$ (3.3) is just that fact written down — a shape sliding right and a shape sliding left, each at speed $c$ (refresher: characteristics →). From it come two questions, mirror images of each other.

Where can a disturbance get to? Suppose something happens at a single place at one instant — a spark at the origin. A time $t$ later the news has travelled a distance $ct$ in each direction and no further, so it can be felt only within $|x| \le ct$. As $t$ grows that set grows: a widening interval in one dimension, a growing disk in two, an expanding sphere in three. Swept out over all time it is the region of influence. Outside it, as far as the target is concerned, the event has not happened yet.

Where could the field here-and-now have come from? Turn it around. To know the field at one chosen place and time — a target $(x_\star, t_\star)$ — what must we know about the starting conditions? A disturbance that began a distance $d$ away takes a time $d/c$ to arrive, so by the time $t_\star$ only starting points within $d = c\,t_\star$ of the target have had time to reach it. The field at the target is fixed entirely by the initial data on the interval $|x - x_\star| \le c\,t_\star$, and not at all by anything outside it. That interval is the target’s domain of dependence.

Two features of that interval are where intuition usually slips, so they are worth stating plainly:

• It is an interval, not a point. One might guess the field at a place depends only on what was happening at that same place to begin with. It does not — the wave carries news inward from both sides, so the value at the target is assembled from all the initial data within $c\,t_\star$ of it.
• Its width is set by how long you wait, and it is never all of space. The interval has width $2 c\,t_\star$. A target just after the start ($t_\star$ small) needs only a sliver of initial data around it; a target far in the future needs a wider stretch, because the wave has had longer to bring in news from afar. As $t_\star \to 0$ the interval shrinks to the single point beneath the target — a point depends only on itself — and it fans outward from there. Only after infinite time would it cover everything. (So in the diagram below, the lit band at $t = 0$ is not “all $x$”; it is this finite interval, drawn against the full initial profile to show how much of it actually matters.)

The two questions are one relationship read in opposite directions: a starting point $x_0$ can influence a target $(x_\star, t_\star)$ exactly when $|x_\star - x_0| \le c\,t_\star$ — equivalently, the target lies in $x_0$‘s region of influence, and $x_0$ lies in the target’s domain of dependence. The $(x,t)$ diagram is the bookkeeping of that relationship: signals run along the slanted characteristics $x \pm ct = \text{const}$, and the cone — a triangle in one dimension — is the set of all initial points wired to the target.

Three views of one event, side by side. The map shows the wavefront as an expanding ring of radius $ct$ — the picture a textbook draws but cannot animate; the profile is the actual wave along a cut through that ring; the spacetime plot stacks every profile into the light cone. The link to hold onto: the ring meets the cut at $x = \pm ct$, and those two points are simultaneously the two pulses in the profile and the two edges of the cone. Toggle to domain of dependence to read it backward — the disk of data that can reach a target. Either way the boundary moves at exactly $c$. (One consequence is hiding here: every disturbance moves at the same $c$, so a 1-D pulse keeps its shape — the wave equation is non-dispersive, a fact that fails only when a medium, geometry, or nonlinearity intervenes.)

The “cone” is only a wedge because the picture above is one-dimensional. Add a second spatial dimension and it becomes a genuine cone: a point event radiates an expanding circle, and stacking that circle along the time axis is the 3-D light cone itself.

The cross-section of the cone at any instant is exactly the wavefront circle beside it. (In true three-dimensional space the wavefront is an expanding sphere $r = ct$ and the cone gains a fourth dimension we can no longer draw — but the rule is the same.)

3. Solutions add

The equation is linear: if $p'_1$ and $p'_2$ solve it, so does $a\,p'_1 + b\,p'_2$. Interference is therefore not a force but bookkeeping — waves pass through one another and simply sum where they overlap. The sharpest consequence: a standing wave is two equal travelling waves moving oppositely,

Equal and opposite, the sum has fixed nodes — a standing wave (3.5). Tip the balance and a travelling component returns. Standing and travelling are one object in different proportions.

4. Every wave is a sum of pure tones

Linearity pays a second dividend. The complex exponentials $e^{i(\mathbf{k}\cdot\mathbf{r}-\omega t)}$ with $\omega = c|\mathbf{k}|$ are the eigenfunctions of the equation: each evolves on its own, merely acquiring a phase. So any field — speech, a handclap, an orchestra — is a superposition of these modes (Foundations 7.1), and propagating an arbitrary signal reduces to propagating its spectrum. The Fourier basis diagonalises the wave equation, exactly as the spectral theorem diagonalises a self-adjoint operator.

Add harmonics one at a time and an arbitrary target shape assembles from pure sinusoids — the operational content of “any wave is a sum of tones,” made into the whole frequency picture in chapter 8.

5. Boundaries select a spectrum

In free space every wavelength is allowed; the spectrum is continuous. Confine the wave — a string fixed at both ends, a tube, a room, a drumhead — and the boundary conditions admit only a discrete ladder of modes, for the simplest 1-D case

The object chooses its own spectrum: this is why a tube has a pitch and a room has a colour (chapter 7). Resonance is the same fact seen dynamically — drive near an $\omega_n$ and energy piles into that mode (2.4).

Open or close each end of the tube and the allowed modes — and the pitch — jump to a new ladder set entirely by the boundary conditions (7.6).

6. The wave carries energy, split two ways

A wave transports energy, held half in motion and half in compression:

The two halves trade back and forth as the wave oscillates, and in the lossless equation their total is conserved and flows at speed $c$ (chapter 5).

Kinetic and strain energy pass through zero together and peak together; their running total is flat. Energy is the conserved currency the rest of chapter 5 spends.

7. Impedance decides what reflects

Where two media meet, what crosses and what bounces back is set by one number per medium — the specific acoustic impedance $Z = \rho_0 c$. The pressure reflection coefficient is

Matched impedances ($Z_1 = Z_2$) transmit everything; a large mismatch returns nearly all of it. Echoes, anechoic foams, and the middle ear’s job are all this one ratio (5.4, 7.1).

Slide the second medium’s impedance: at a match the wave passes whole; widen the gap and a reflected wave grows, flipping sign as $Z_2$ crosses $Z_1$.

8. Every source is a sum of point responses

Everything above is the free field. A source adds a term to the right-hand side, and the device that solves the forced equation is the Green’s function — the field of a single point impulse. Linearity then gives the field of any source as a weighted sum of impulse responses: a convolution of the source with the Green’s function. Monopole, dipole, and piston are all this one idea (6.6).

Slide one signal across the other and the running overlap traces the output: the source, smeared by the medium’s impulse response, is the field it radiates.

Summary

Eight readings, one equation. Stripped to its load-bearing ideas:

with the division of labour $c \to \text{medium}$, $\nabla^2 \to \text{space}$, boundary conditions $\to \text{object}$. The canonical order in which the book builds on these:

1. Travelling waves — the raw solution (3.3).
2. Fourier — arbitrary waves as sums of modes (chapter 8).
3. Boundaries and modes — the discrete spectrum a bounded object allows (chapter 7).
4. Energy — the conserved currency the wave carries (chapter 5).
5. Impedance — how that energy divides at a junction (5.4).
6. Forcing — how sources launch waves in the first place (chapter 6).
7. Dispersion and damping — where the ideal picture finally bends (chapter 10).

Each later complication — geometric spreading, forcing, damping, dispersion, a moving medium (chapter 9), the failure of linearity at large amplitude (10.4) — bends one bone of this skeleton without breaking it. Hold the skeleton in mind and the rest of the book is variations on a theme.