06. The Inverse-Square Law

Free Field Dispersion Simulator

Mutate raw acoustic energy output vs boundary limits to track how intensity values deteriorate quadratically over distance parameters.

Source Power ($P$): 1.00 W

Observer Radius ($r$): 1.0 m

Spherical Expansion Boundary Metrics (Blue Source / Red Observer Tracking Dot)

Geometric Surface Area

12.57 m²

$A = 4\pi r^2$ expansion boundaries

Energy Intensity Density

79.58 mW/m²

Concentration profile at user distance

Relative Field Attenuation

-6.02 dB

Loss compared to reference $r_0 = 1\text{m}$

The Layman Breakdown: Acoustic Dilution Reality Check

In our daily listening environments, sound bounces off of boundaries. But the Inverse-Square Law assumes a completely open space (like outer space or an open desert field) where sound waves can travel infinitely outward without ever hitting a wall.

🏜️

In an Anechoic Free Field

Move from 1 meter to 2 meters away from your studio monitors, and you lose 6dB instantly. Move to 4 meters, you lose another 6dB.

Sound drops off fast, clean, and predictably because the energy is allowed to escape away from you forever.

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In a Standard Listening Room

Sound waves slam directly into walls, ceilings, and floors, reflecting energy right back into your listening sweet spot.

This trapped reinforcement completely breaks the clean geometric drop, stabilizing volume levels far away inside what is known as the Diffuse Field.

The Physics Engine

The Mathematical Architecture of Dispersion

When a point source projects acoustic energy into space, it radiates as an expanding omnidirectional sphere. Because energy must remain conserved, the total acoustic output power ($P$) spreads across a surface area that expands quadratically:

$$I = \frac{P}{4\pi r^2}$$

Applying logarithmic ratios to compute human decibel perception parameters allows us to mathematically derive the universal "6dB drop per doubling of distance" law profile:

$$\Delta L = 10 \log_{10}\left(\frac{I_2}{I_1}\right) = 20 \log_{10}\left(\frac{r_1}{r_2}\right)$$\n $$\Delta L = 20 \log_{10}\left(\frac{1}{2}\right) \approx -6.02\text{ dB}$$

$P$

Source Power: The absolute acoustic energy output capacity generated directly at the point source core.

$r$

Radius: The physical separation distance vector. As $r$ doubles, the boundary surface area increases by a factor of 4 ($2^2$), stretching the energy thinner.

$\Delta L$

Logarithmic Level Delta: The resulting drop calculated in Decibels. Every single exact doubling of physical distance in a true free field results in an absolute drop of 6.02 dB in sound pressure level.