Inverse Square Law

Reference data and engineering information about inverse square law for miscellaneous applications.

inversesquarelaw

Overview

Engineering reference data for Inverse Square Law in miscellaneous.

Key Formulas

Unit Conversion

$y = x \cdot k$

Multiply by conversion factor.

Linear Interpolation

$y = y_1 + \frac{(x - x_1)(y_2 - y_1)}{x_2 - x_1}$

Estimate between two known points.

Percentage

$p = \frac{\text{part}}{\text{whole}} \times 100\%$

Part as fraction of whole.

Variables

Symbol	Description	Unit
$x$	Input value	—
$y$	Output value	—
$k$	Conversion factor	—

Practical Applications & Examples

The inverse square law provides critical predictions for sound attenuation in unobstructed environments. Below are practical calculations and reference data.

Sound Propagation Data Table

The following table shows how sound pressure level (Lp) decreases with distance for a point source in a free field, consistent with a 6 dB reduction per distance doubling.

13 rows

Sound pressure level at various distances from a source, illustrating the inverse square law.
Distance(ft)	Sound Pressure Level (Lp)(dB)
1.25	134
2.5	128
5	122
10	116
20	110
40	104
80	98
160	92
320	86
640	78
1280	74
2560	68
5120	62

Source: engineeringtoolbox.com

Calculation Examples

Example 1: Rifle Shot Sound Attenuation Given a measured sound pressure level ( $L_{p1}$ ) of 134 dB at 1.25 feet, find the level at 80 feet.

Calculate the difference ( $\Delta L$ ) using the formula: $\Delta L = 20 \log\left(\frac{R_2}{R_1}\right) = 20 \log\left(\frac{80}{1.25}\right) = 20 \log(64) \approx 36 \text{ dB}$
Apply the difference to find $L_{p2}$ : $L_{p2} = L_{p1} - \Delta L = 134 \text{ dB} - 36 \text{ dB} = 98 \text{ dB}$

Example 2: Industrial Noise Assessment A machine has a sound pressure level ( $L_{p1}$ ) of 110 dB at 1 meter. Assess the level in a working area 5 meters away.

Calculate the reduction ( $\Delta L$ ): $\Delta L = 20 \log\left(\frac{5}{1}\right) = 20 \log(5) \approx 14 \text{ dB}$
Determine the new sound level ( $L_{p2}$ ): $L_{p2} = 110 \text{ dB} - 14 \text{ dB} = 96 \text{ dB}$ Note: A level of 96 dB often exceeds permissible exposure limits for continuous work, necessitating engineering controls like barriers or enclosures.