4.5
What is sound?
§4 · The acoustic wave equation

4.5 Linearisation and the fluid-mechanics wave equation

We have, after three lessons, a closed linear system:

∂tρ′+ρ0∇⋅v′  =  0(continuity)\partial_t \rho' + \rho_0 \nabla \cdot \mathbf{v}' \;=\; 0 \qquad \text{(continuity)} ρ0∂tv′  =  −∇p′(Euler)\rho_0 \partial_t \mathbf{v}' \;=\; -\nabla p' \qquad \text{(Euler)} p′  =  c2 ρ′(adiabatic equation of state)p' \;=\; c^2\, \rho' \qquad \text{(adiabatic equation of state)}

Three equations, three perturbation fields (p′p', ρ′\rho', v′\mathbf{v}'). This lesson eliminates two of them and arrives at a single second-order PDE for p′p': the acoustic wave equation.

The combination

Take the time derivative of the continuity equation:

∂t2ρ′+ρ0∇⋅(∂tv′)  =  0.\partial_t^2 \rho' + \rho_0 \nabla \cdot (\partial_t \mathbf{v}') \;=\; 0.

From Euler, ∂tv′=−(1/ρ0) ∇p′\partial_t \mathbf{v}' = -(1/\rho_0)\, \nabla p'. Substitute:

∂t2ρ′−∇⋅(∇p′)  =  0,\partial_t^2 \rho' - \nabla \cdot (\nabla p') \;=\; 0,

i.e.

∂t2ρ′  =  ∇2p′.\partial_t^2 \rho' \;=\; \nabla^2 p'.

Use the equation of state p′=c2ρ′p' = c^2 \rho' to replace ρ′\rho' with p′/c2p'/c^2 on the left:

    ∂2p′∂t2  =  c2 ∇2p′.    \boxed{\;\;\frac{\partial^2 p'}{\partial t^2} \;=\; c^2\, \nabla^2 p'.\;\;}

The acoustic wave equation. Second-order in time and in space, linear in p′p', with the adiabatic sound speed cc as the only parameter.

▶ Why we could pick *any* of $p'$, $\rho'$, or velocity potential

The same three equations give wave equations for every perturbation field, because they are linearly related.

For ρ′\rho': from ∂t2ρ′=∇2p′\partial_t^2 \rho' = \nabla^2 p' and p′=c2ρ′p' = c^2 \rho', ∂t2ρ′=c2∇2ρ′\partial_t^2 \rho' = c^2 \nabla^2 \rho'.

For v′\mathbf{v}': take the time derivative of Euler, ρ0∂t2v′=−∇∂tp′\rho_0 \partial_t^2 \mathbf{v}' = -\nabla \partial_t p'. Use continuity (after multiplying by c2c^2) for ∂tp′=−c2ρ0∇⋅v′\partial_t p' = -c^2 \rho_0 \nabla \cdot \mathbf{v}', so ρ0∂t2v′=c2ρ0∇(∇⋅v′)\rho_0 \partial_t^2 \mathbf{v}' = c^2 \rho_0 \nabla (\nabla \cdot \mathbf{v}'). For an irrotational flow (∇×v′=0\nabla \times \mathbf{v}' = 0), ∇(∇⋅v′)=∇2v′\nabla(\nabla \cdot \mathbf{v}') = \nabla^2 \mathbf{v}', giving ∂t2v′=c2∇2v′\partial_t^2 \mathbf{v}' = c^2 \nabla^2 \mathbf{v}'.

For a velocity potential ϕ\phi with v′=∇ϕ\mathbf{v}' = \nabla \phi: from Euler, ρ0∂t∇ϕ=−∇p′\rho_0 \partial_t \nabla \phi = -\nabla p', so p′=−ρ0∂tϕp' = -\rho_0 \partial_t \phi (up to a function of time that’s set to zero). Substituting into the wave equation for p′p' gives ∂t2ϕ=c2∇2ϕ\partial_t^2 \phi = c^2 \nabla^2 \phi.

Same equation; different field. Which to use is convenience. Pressure is what microphones measure and what our intuitions are built on. Velocity is what’s needed to compute intensity and momentum. Velocity potential is the cleanest for radiation problems. We will move between them as needed.

What we have built

We started from three physical laws, each of them obvious individually:

We linearised about equilibrium — three steps, each throwing away terms quadratic in the perturbations — and combined. Out fell a single PDE governing every linear-acoustic phenomenon. The form

∂t2p′  =  c2∇2p′\partial_t^2 p' \;=\; c^2 \nabla^2 p'

is not negotiable: it is what the three laws say, in their small-perturbation limit. We did not assume oscillations. We did not assume waves. The wave equation is the small-perturbation content of fluid mechanics, and we have just derived it.

Visualising what propagates

molecules at equilibrium ± local displacement u(x, t)pressure p(x, t) = −ρ₀ c² · ∂u/∂xposition x →
120 px
±7 px
direction:

The renderer shows the molecular displacement (top) and the pressure perturbation (bottom). Note that they are a quarter-wavelength out of phase: where the displacement is maximum, the pressure perturbation is zero; where the displacement gradient is maximum (which is also where molecules are bunched together), the pressure is at its peak. Toggle between left-going, right-going, and standing-wave modes; the standing case is what we’d see between two reflecting boundaries.

Three more routes, four altogether

The wave equation is real. It is the spine of acoustics. But its inevitability is not yet obvious — it could feel like the wave equation happens to be what falls out of these three particular equations, and a different starting point might have given something else.

The next three lessons argue the opposite. From a 1-D chain of mass-springs (lesson 4.6), from a kinetic-theory treatment of pressure as molecular momentum flux (4.7), and from a Lagrangian variational principle (4.8), we will derive the same wave equation. After all four derivations there will be no doubt: this is the equation that any continuous, small-perturbation, restoring medium obeys.