The inverse square law — and when it doesn't apply

Derivation

A point source radiating uniformly in all directions produces a spherical wavefront. The acoustic power P radiated by the source is constant; it spreads across an ever-larger spherical surface as the wave expands. The surface area of a sphere of radius r is 4πr². Sound intensity I — power per unit area — is therefore:

I = P / (4πr²)

Since intensity is proportional to the square of pressure, and pressure is proportional to 1/r:

I ∝ 1/r² p ∝ 1/r

Doubling the distance r reduces intensity by a factor of 4 (−6 dB) and pressure by a factor of 2 (also −6 dB, since SPL uses a factor of 20 in the logarithm). This 6 dB-per-doubling relationship is the inverse square law.

In terms of SPL, if the level at distance r₁ is known:

L₂ = L₁ − 20 × log₁₀(r₂ / r₁) dB SPL

For a doubling of distance (r₂ = 2r₁): L₂ = L₁ − 20 × log₁₀(2) ≈ L₁ − 6 dB.

When it applies

The inverse square law holds under two conditions:

1. Free field. There are no reflecting surfaces. The wave expands without interruption. In practice, this means an anechoic chamber or an outdoor environment away from the ground and other reflective surfaces.
2. Far field. The measurement point is far enough from the source that the source can be treated as a point — typically several source dimensions away. In the near field, the wavefront is not yet spherical and the relationship between pressure and distance is more complex.

When it does not apply

Reverberant field. In a real room, reflected energy accumulates to form a diffuse reverberant field whose level is approximately uniform throughout the room and independent of distance from the source. At distances beyond the critical distance — the point where direct and reverberant energy are equal — the total SPL changes little with increasing distance. The inverse square law predicts a continuous fall in level with distance; in a reverberant room, the level plateau is dominated by reflected energy. Critical distance and its practical implications covers this in detail.

Half-space radiation (2π field). A source mounted flush in a large flat surface — a loudspeaker in a wall, for example — radiates into a hemisphere rather than a full sphere. The surface area is 2πr² rather than 4πr², so intensity at a given distance is doubled (+3 dB) relative to free-field radiation. The 6 dB-per-doubling rule still applies with distance, but the absolute level is higher. This is the 2π condition; radiation into full free space is the 4π condition. Loudspeaker datasheets may specify sensitivity in either condition; the distinction matters when comparing drivers.

Line sources. An infinitely long line source — approximated by a column loudspeaker, a road, or a train — radiates cylindrically rather than spherically. Intensity falls as 1/r (not 1/r²), so level falls at 3 dB per doubling of distance, not 6 dB. The inverse square law does not apply. In practice, real line sources transition from cylindrical spreading (3 dB/doubling) close to the source to spherical spreading (6 dB/doubling) at distances large compared to the source length.

Near field. Very close to a source, the wavefront is not yet spherical and the field contains reactive (non-propagating) energy. Pressure can vary non-monotonically with distance, and the 6 dB rule does not hold.

Large distributed sources. An extended source — a large panel, a room surface, an HVAC duct — cannot be treated as a point. Level falls more slowly than 6 dB/doubling at distances comparable to the source dimensions, transitioning to inverse-square behaviour only in the far field.

Practical implications

When measuring a loudspeaker in a room, the inverse square law holds only in the direct field close to the driver — and only if the measurement is taken before significant reflections arrive. Window the impulse response to exclude reflections, or take measurements outdoors. Do not assume that level at 1 m predicts level at 2 m using the inverse square law unless the measurement conditions genuinely approximate free field.

In room acoustic design, the transition from direct-field (inverse square law) behaviour to reverberant-field behaviour happens at the critical distance. For a typical small room with moderate treatment, this distance is often only 1–2 m. Beyond it, increasing the source power raises both direct and reverberant levels equally, with no change in the direct-to-reverberant ratio.