1.5 The question for the rest of the book

We have arrived at a sharp question. Air at equilibrium has fixed pressure $p_0$ , density $\rho_0$ , zero average velocity. A sound is a small coherent perturbation $(p', \rho', \mathbf{v}')$ on top of that equilibrium. The perturbation propagates. What is the equation that governs that propagation?

The answer — which we will spend the next three chapters deriving — is the acoustic wave equation:

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

Same form everywhere. Two spatial derivatives, two time derivatives, a constant $c^2$ between them. The constant $c$ turns out to be the speed of sound. The equation governs every linear-acoustic phenomenon you will meet in the book: how plane waves propagate, how they reflect, how they superpose into modes of a tube, how they radiate from a vibrating source, how they get bent by a temperature gradient or refracted at a flow boundary.

What we still owe the reader

This equation has not yet been derived from anything. We wrote it down to fix the destination. The work of the next three chapters is to arrive at it from first principles — four different times, by four different routes:

From fluid mechanics. Newton’s second law for a fluid slab, conservation of mass, an equation of state — each linearised about equilibrium, combined.
From a lattice of oscillators. A chain of mass-springs in the limit of large $N$ and small spacing.
From kinetic theory. Pressure as a flux of molecular momentum; sound as a local imbalance in that flux.
From a variational principle. A Lagrangian density for the perturbation field, with the wave equation as the stationary action.

Each route reveals a different facet of why the wave equation has the form it has — and, satisfyingly, all four agree on the value of $c$ .

Before we can do that

Before we can derive the wave equation from fluid mechanics, we need:

The mechanics of a single oscillator (mass-spring, damped, driven). This is chapter 2.
The behaviour of coupled oscillators in the continuum limit — the simplest wave: a vibrating string. This is chapter 3.

The first gives us the conceptual scaffolding (oscillation, resonance, the complex-exponential method). The second gives us our first wave equation in a setting where we can see the displacement of the medium. Then, in chapter 4, we will be ready to derive the fluid acoustic wave equation four different ways, and the rest of the book becomes an exploration of its consequences.

The journey starts with one mass on one spring.