4.10 Reading the solutions — what the wave equation does

We have the equation, four times over:

\frac{\partial^2 p'}{\partial t^2} \;=\; c^2 \nabla^2 p'.

Deriving it tells us why it holds. It does not tell us how to read it — what its solutions can and cannot do, and which of its features is responsible for which acoustic fact. That reading is the subject of this lesson. Nothing new is derived; instead each property of the solutions is stated plainly and pointed to the later lesson that develops it. Treat this as the map for chapters 5 through 10.

The equation, term by term

The left side is an acceleration: the second time-derivative of the pressure field at a point. The right side is a curvature: $\nabla^2 p'$ measures how much $p'$ at a point differs from the average of its immediate neighbours. The equation says these are proportional. A point accelerates because its neighbours are out of line with it. Where the field is locally concave — a pressure valley — the curvature is positive and the point accelerates upward, back toward its neighbours. The wave equation is local restoring geometry, the continuum version of a mass pulled by the springs on either side.

Three quantities sit in the equation, and each owns a different part of the physics:

$c$ contains the medium — through $c^2 = (\partial p / \partial \rho)_s$ , every thermodynamic fact about the gas is compressed into one number (lesson 4.9).
$\nabla^2$ contains space — the dimensionality and coordinates of the region the wave lives in.
The boundary conditions, not yet written, contain the object — the string, the tube, the room, the cochlea.

Change the medium and only $c$ changes; change the object and only the boundary conditions change. This separation is why the same equation serves a violin string and a concert hall.

▶ Curvature causes acceleration: reading $\nabla^2$ as a neighbour-difference Derivation

Discretise the 1-D equation $\partial_t^2 p' = c^2 \partial_x^2 p'$ on a grid of spacing $\Delta x$ . The second spatial derivative is the standard three-point difference

\partial_x^2 p' \;\approx\; \frac{p'_{i+1} - 2 p'_i + p'_{i-1}}{\Delta x^2}.

The numerator is exactly twice the gap between $p'_i$ and the average of its two neighbours, $\tfrac12(p'_{i+1} + p'_{i-1})$ . So the equation reads

\ddot p'_i \;=\; \frac{c^2}{\Delta x^2}\Big[\, \underbrace{\tfrac12(p'_{i+1} + p'_{i-1})}_{\text{neighbour average}} - p'_i \,\Big]\cdot 2.

If a point lags below the average of its neighbours, it accelerates upward; if it bulges above, it accelerates back down. The Laplacian is the multidimensional version of the same statement, comparing each point to the average over a small surrounding sphere. This is the microscopic content of “curvature causes acceleration,” and the same difference is what a finite-difference solver computes (Foundations 9.3).

A disturbance travels, and at finite speed

In one dimension the general solution is d’Alembert’s (3.3):

p'(x, t) \;=\; f(x - c t) + g(x + c t),

a right-going profile plus a left-going one, each rigid. Two consequences are built into this form and survive into higher dimensions (refresher: characteristics and the domain of dependence →):

Finite speed. The value at $(x, t)$ depends only on initial data within the domain of dependence $[x - ct,\, x + ct]$ . Nothing outside that interval can have reached the point yet. In three dimensions a flash at the origin is felt only on the expanding cone $r = ct$ .
No dispersion. Because every disturbance moves at the single speed $c$ , a 1-D pulse keeps its shape exactly. Shape is preserved unless boundaries, geometry, a frequency-dependent medium, or nonlinearity intervene — the four ways the book later complicates this picture.

view:

wave speed c: 1.00

Drag the black dot anywhere in the plot. Two characteristics — one left-going (red), one right-going (blue) — reach back from (x, t) to the t = 0 axis. The bold gold segment between them is the domain of dependence: the only stretch of initial data that can possibly affect the solution at (x, t). Everything outside that segment is causally disconnected.

Drag the target point: the two characteristics $x \pm ct = \text{const}$ that reach it strike $t = 0$ at the ends of its domain of dependence. Slide $c$ and the cones widen or narrow — information travels faster or slower, but always at a strict, finite limit.

Solutions add

The equation is linear: if $p'_1$ and $p'_2$ are solutions, so is $a\,p'_1 + b\,p'_2$ . Interference is therefore not a new force — it is bookkeeping. Two waves crossing pass through one another and emerge unchanged; where they overlap, their pressures simply sum. The most important special case is that a standing wave is two equal travelling waves moving oppositely (3.5):

\sin(kx)\cos(\omega t) \;=\; \tfrac12\big[\sin(kx - \omega t) + \sin(kx + \omega t)\big].

Travelling and standing are not two kinds of wave but two views of the same object.

input frequency f

1 kHz

characteristic place: 15.2 mm from stapes
Q (fixed): 8

Every wave is a sum of pure tones

Linearity has a deeper payoff. The complex exponentials

p' \;=\; A\, e^{i(\mathbf{k}\cdot\mathbf{r} - \omega t)}, \qquad \omega = c\,|\mathbf{k}|,

are the eigenfunctions of the wave equation: each one evolves independently, multiplied by a phase, and the dispersion relation $\omega = c k$ holds for every one of them. Any field — speech, a handclap, an orchestra — is a superposition of these modes (Foundations 7.1, 7.2), and because each evolves on its own, propagating an arbitrary signal reduces to propagating its spectrum. This is the entire programme of the frequency picture (chapter 8). The Fourier basis diagonalises the wave equation in the same sense the spectral theorem diagonalises a self-adjoint matrix.

Boundaries pick the spectrum

In free space the allowed wavelengths form a continuum. Confine the wave — a string clamped at both ends, a tube, a rectangular room, a membrane, a cochlea — and the boundary conditions admit only a discrete set of modes with eigenfrequencies

\omega_n \;=\; \frac{n\pi c}{L} \qquad (\text{string or tube of length } L),

and their multidimensional generalisations (chapter 7). The object selects its own spectrum; this is why a tube has a pitch and a room has a colour. Resonance is the dynamic face of the same fact: drive a bounded system near one of its $\omega_n$ and energy accumulates selectively in that mode (2.4, 8.4).

The field carries energy and momentum

A sound wave is not just a shape that moves; it transports energy. The energy density splits into a kinetic part and a strain (gradient) part,

\mathcal{E} \;=\; \tfrac12 \rho_0 |\mathbf{v}'|^2 \;+\; \tfrac{1}{2\rho_0 c^2}\, p'^2,

and the two trade off as the wave oscillates, with the total conserved in the lossless equation (chapter 5). The flow of that energy is the intensity, and the quantity that governs how a wave divides at an interface is the impedance $Z = \rho_0 c$ : the reflection coefficient at a junction is

R \;=\; \frac{Z_2 - Z_1}{Z_2 + Z_1},

so mismatch makes echoes and matching makes transmission (5.4, 7.1).

Any source is a sum of point responses

Everything above concerns the free field. Sources enter as a term on the right-hand side, and the device that solves the forced equation is the Green’s function: the field produced by a single point impulse. Because the equation is linear, the response to any source whatever is a weighted sum of point-impulse responses — a convolution of the source with the Green’s function. The monopole, dipole, and piston of chapter 6 are all special cases of this one idea, developed in 6.6.

The compression

Strip the book to its load-bearing ideas and the wave equation reads:

\boxed{\;\;\text{wave equation} \;=\; \text{local curvature} \;+\; \text{finite speed} \;+\; \text{modal decomposition} \;+\; \text{boundary-selected spectrum}\;\;}

with the division of labour $c \to \text{medium}$ , $\nabla^2 \to \text{space}$ , boundary conditions $\to \text{object}$ . Everything the rest of the book adds is a modification of this skeleton: geometric spreading in higher dimensions (chapter 6), forcing by sources (chapter 6), damping and dispersion (chapter 10), motion of the medium (chapter 9), and the breakdown of linearity itself at large amplitude (10.4). None of these changes the skeleton; each bends one of its bones.

The path through the rest of the book

The canonical order in which the ideas build on one another:

Travelling waves — the raw solution (3.3).
Fourier — arbitrary waves as sums of modes (chapter 8).
Boundaries — what happens when the wave meets something (chapter 7).
Modes — the discrete spectrum a bounded object allows (3.5, 7.6).
Energy — what the wave carries (chapter 5).
Impedance — how it divides at boundaries (5.4).
Forcing — how sources launch it (chapter 6).
Dispersion and damping — where the ideal picture finally bends (chapter 10).

⏳ The history — Rayleigh's synthesis

By the 1870s the pieces of this reading existed in scattered form — d’Alembert’s travelling waves, Bernoulli’s and Fourier’s modal sums, George Green’s impulse responses, Helmholtz’s resonators. It was John William Strutt, third Baron Rayleigh, who assembled them into a single coherent theory in The Theory of Sound (two volumes, 1877–78; Rayleigh 1894). Rayleigh’s organising move was exactly the one of this lesson: treat the wave equation as a linear operator, expand every problem in its normal modes, and read energy, radiation, and resonance off the expansion. The book is still in print and still cited; much of the vocabulary of acoustics — “normal mode,” the Rayleigh quotient for estimating frequencies, the Rayleigh integral for radiation — descends directly from it.