Derive the linear acoustic equations

Consider a fluid initially at rest, with constant equilibrium density and pressure. Write its density as an equilibrium value plus a small perturbation, and call its small longitudinal particle velocity u. A short fixed interval gains mass when more fluid enters than leaves. At first order, the difference between these boundary flows is equilibrium density times the spatial derivative of u. Uniform translation therefore changes no density; a velocity that decreases along the interval compresses it.

The exact continuity equation contains the product of density perturbation and velocity. Linearization drops that product because both factors are already small. This is an ordering argument, not a claim that products never matter. Perturbation density divided by equilibrium density, and particle speed divided by sound speed, must remain much less than one. The equilibrium must also vary slowly enough that treating it as uniform over the region of interest is justified.

A fluid element feels pressure forces from both ends. Equal pressures cancel; a pressure that decreases to the right pushes the element rightward. Dividing the net force by its mass gives the linear momentum equation. The pressure itself can be large while its perturbation is small. A uniform atmospheric background supplies no net acceleration in this idealized horizontal problem. Ignoring viscosity and external forcing makes the equation conservative, not universally exact for real sound.

Close the pair of balance laws with a constitutive relation between pressure and density perturbations. For reversible, locally adiabatic changes, the proportionality is the isentropic derivative of pressure with respect to density. Differentiate continuity in time and substitute the momentum equation to obtain a wave equation. Its speed depends on material stiffness relative to inertia. Conservation alone does not specify that stiffness; an equation of state and the thermal regime are essential inputs.

A right-moving sinusoidal pressure disturbance has phase kx−ωt and satisfies ω=ck in this nondispersive model. A fixed crest advances at c while individual fluid elements oscillate about their equilibrium positions to first order. Their excursions are not the distance traveled by the crest. Mean streaming and net mass transport, when present in real experiments, require higher-order effects or boundaries beyond this first-order description. Do not infer permanent transport simply from a moving pressure maximum.

Use the two-wave lab to test linear superposition. Before moving the phase control, predict what equal amplitudes and a half-cycle phase difference will do to the sum. Then compare with a quarter-cycle offset. The display contains two same-direction sinusoids, not a density field or a compressible-fluid solver. Its exact cancellation illustrates linear addition; it cannot reveal viscosity, shocks, thermal losses, or the opposite-direction motion needed for standing sound in a pipe.