1.4 What a sound is — a small departure from equilibrium

Equilibrium air has a uniform pressure $p_0$, uniform density $\rho_0$, and zero average flow velocity. A sound, anywhere in that air, is a tiny coherent departure from those equilibrium values. We write it as a perturbation around the equilibrium state:

The primed quantities $p'$, $\rho'$, $\mathbf{v}'$ are the sound field: the deviations from equilibrium. They are small. For conversational speech the pressure perturbation $p'$ is about $10^{-2}$ Pa — seven orders of magnitude smaller than the equilibrium $p_0 \approx 10^5$ Pa. For very loud sounds we might see a few hundred Pa. For something painfully loud — a jet engine at close range — perhaps a few thousand. We almost never approach $p_0$ itself; if we did, the linear theory we are about to build would break down, and the air would be in the strongly nonlinear regime of shocks and cavitation — the physics of chapter 10, not linear acoustics.

Longitudinal: along the direction of travel

The pressure fluctuation is longitudinal: the molecules move back and forth along the direction the wave is travelling, not perpendicular to it like a wave on water. Where the fluctuation is positive (above ambient), the molecules are slightly compressed — closer together. Where it is negative, they are slightly rarefied.

The pattern propagates because compressed regions push neighbouring regions, and rarefied regions pull. The molecules themselves do not travel any net distance; the pattern of compression does. This is the central physical picture of an acoustic wave.

Three perturbation fields, one phenomenon

We wrote three perturbation fields — pressure $p'$, density $\rho'$, velocity $\mathbf v'$ — but they are not independent. Three fluid-mechanics laws relate them: conservation of mass, Newton’s second law, and an equation of state (the subject of chapter 4). Together these make any one field determine the other two, so a sound carries one degree of freedom, not three. For most of the book that field is the pressure perturbation $p'$.

A few chapters instead use a single scalar, the velocity potential $\phi$, defined so that $\mathbf v' = \nabla\phi$. This is allowed because acoustic flow in an ideal fluid is irrotational — it has no local swirl, $\nabla\times\mathbf v' = 0$ — and any curl-free field is the gradient of some scalar, the converse of the identity $\nabla\times\nabla\phi = 0$ (refresher: divergence and curl →). The potential packs the three components of $\mathbf v'$ into one function, from which $p'$ and $\rho'$ follow, so the wave equation can be solved for the single unknown $\phi$. The two pictures are equivalent; the book uses whichever is more convenient.

Smallness and linearity

Calling $p'$ small relative to $p_0$ is the assumption that lets us linearise the fluid equations: expand them around equilibrium and keep only first order in the primed quantities, discarding products like $\rho'\mathbf v'$ that are second-order small (refresher: linearisation →). What survives is a linear PDE — the wave equation — and linearity is what makes superposition hold: two sounds pass through each other unchanged, and any signal can be built from independent Fourier components. Most of acoustics is tractable for that reason.

The smallness is a genuine physical approximation, and every “small perturbation” in the book is shorthand for dropping the terms quadratic in the primed quantities. Chapter 10 restores them and recovers the nonlinear corrections — shock formation, streaming, the entry to cavitation — where the linear theory fails. Until then, the perturbations are small, the object is defined, and we can derive its equation.