Sound waves — pressure, particle velocity, and wave propagation

What a sound wave is

Sound propagates through a medium — air, water, or a solid — as a mechanical wave. In air, a sound wave is a travelling pattern of pressure variations above and below the ambient atmospheric pressure. These pressure variations cause the air molecules to move back and forth in the direction of wave travel, a motion described as longitudinal.

This is distinct from transverse waves (such as waves on a string), in which the displacement is perpendicular to the direction of propagation. Sound in fluids is always longitudinal.

Compression and rarefaction

As a sound wave passes through air, alternating regions of compression (above-ambient pressure) and rarefaction (below-ambient pressure) move through the medium. The air molecules themselves do not travel with the wave — they oscillate about their rest positions. It is the pattern of pressure variation that propagates, not the medium itself.

The amplitude of these pressure variations is what we measure as sound pressure. At 1 kHz and 0 dB SPL, the peak pressure variation is approximately 28 μPa — about one quarter-billionth of atmospheric pressure. The ear is extraordinarily sensitive.

Particle velocity

Alongside pressure, sound waves involve particle velocity — the oscillatory velocity of the air molecules as they move back and forth. Particle velocity (symbol: u) is a vector quantity with both magnitude and direction; sound pressure (symbol: p) is a scalar.

In a free field — away from boundaries and obstacles — pressure and particle velocity are in phase with each other, and their ratio defines the characteristic acoustic impedance of the medium:

Z₀ = p / u = ρ × c

where ρ is the air density (approximately 1.21 kg/m³ at 20°C) and c is the speed of sound. At 20°C:

Z₀ ≈ 1.21 × 343 ≈ 415 Pa·s/m

This quantity, the specific acoustic impedance of air, appears throughout transducer design, acoustic modelling, and the analysis of reflections at boundaries. When a sound wave encounters a boundary between two media with different acoustic impedances, part of the energy is reflected and part is transmitted. The greater the impedance mismatch, the stronger the reflection — which is why sound reflects efficiently from rigid walls (whose impedance far exceeds that of air) but is partially absorbed by materials whose impedance is closer to that of air.

Near the source: reactive and active fields

Close to a sound source — within roughly one wavelength — the pressure and particle velocity are not in phase. In this near field, the relationship between pressure and velocity is complex and reactive: energy oscillates back and forth between the source and the field without being radiated. This stored, non-propagating energy does not contribute to the radiated sound power.

In the far field, beyond approximately one wavelength from the source, the wave behaves as described above: pressure and particle velocity are in phase, impedance is real, and energy propagates outward. The transition between near and far field behaviour has practical consequences for loudspeaker measurement — measurements taken in the near field do not reflect the true far-field frequency response. The near field and far field: acoustic zones explained covers this distinction in detail.

The wave equation

The propagation of sound is governed by the acoustic wave equation, which in one dimension is:

∂²p/∂x² = (1/c²) × ∂²p/∂t²

This relates the spatial curvature of the pressure field to its time evolution. Solutions to this equation in three dimensions describe the behaviour of sound in rooms, around obstacles, through apertures, and from sources of different geometries. Most practical acoustic modelling — whether geometric (ray tracing, image method) or wave-based (finite element analysis) — derives from this equation.

Plane waves and spherical waves

Two idealised wave geometries are particularly useful:

Plane waves — in which wavefronts are flat parallel surfaces and pressure is uniform across each front — arise far from a source or in a tube or duct. The relationship between pressure and particle velocity is simply Z₀ = ρc.

Spherical waves — which radiate outward from a point source in all directions equally — describe the free-field radiation from a small source at distances large compared to the source dimensions. The pressure amplitude of a spherical wave falls as 1/r (a 6 dB reduction per doubling of distance) as the energy spreads across an ever-larger spherical surface. This is the basis of the inverse square law, discussed in The inverse square law — and when it doesn't apply.