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 we would have very different problems to worry about than 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

Notice that we wrote three perturbation fields above — pressure, density, velocity — not one. They are not independent. The wave equation (next chapter onward) will relate them via three fluid-mechanics laws (conservation of mass, Newton’s second law, and an equation of state). The result is that knowing any one of them at all points and at all times determines the other two.

For most of the book we will work with $p'$ (the pressure perturbation) as the canonical field. Some chapters favour the velocity potential $\phi$ defined by $\mathbf{v}' = \nabla \phi$, which exists because acoustic flow is irrotational ($\nabla \times \mathbf{v}' = 0$) in an ideal fluid. The two pictures are equivalent and we will use whichever is more convenient.

”Small” is doing real work

The smallness of $p'$ compared to $p_0$ is not an aesthetic claim. It is the assumption that allows us to linearise the fluid equations. Linearisation throws away terms like $\rho' \mathbf{v}'$ that are second-order small. What remains is a linear PDE — the wave equation — whose general solution is a sum of independent modes that pass through each other without interacting. Linearity is what makes Fourier decomposition useful, makes superposition work, and makes most of acoustics analytically tractable.

Throughout the book, every “small perturbation” is shorthand for throwing out terms quadratic in the primed quantities. When we eventually return to those quadratic terms in chapter 10, we will recover the nonlinear corrections — shock formation, second-order streaming, the entry to cavitation — and the linear theory will visibly start to creak.

For now, small is small. We have defined the object. We are ready to derive its equation.