Inverse-Square Law for Sound Intensity
Determines sound level decay in a free field: sound pressure level drops by exactly 6.02 dB for every doubling of distance from a point source.
\[ I = \frac{P_{sound}}{4\pi r^2} \implies \Delta L_p = 20 \log_{10}\left( \frac{r_1}{r_2} \right) \]
Plain Text Formula:
I = P / (4 * pi * r^2) ==> Delta_Lp = 20 * log10(r1 / r2)
📐 Variable Definitions & Parameters
| Symbol | Variable Name | Standard Unit | Physical Definition |
|---|---|---|---|
| \[ I \] | Sound Intensity | W/m² | Acoustic power flowing through unit area |
| \[ P_{sound} \] | Acoustic Sound Power | Watts (W) | Total acoustic output of the omnidirectional source |
| \[ r \] | Distance from Source | meters (m) | Spherical radius from sound emitter |
| \[ \Delta L_p \] | Sound Level Delta | dB | Change in perceived decibel sound level |
💡 Worked Engineering Example
Problem Statement:
If a speaker produces 94 dB SPL at 1 meter, calculate sound pressure level at 4 meters in a free field.
Step-by-Step Solution:
- Distance quadrupled (r2 / r1 = 4)
- Delta_Lp = 20 * log10(1 / 4) = 20 * (-0.60206) = -12.04 dB
- L2 = 94 - 12.04 = 81.96 dB SPL
🎯 Result: At 4 meters, the sound pressure drops to 82.0 dB SPL.