Equivalent Sound Level

Overview

Engineering reference data for Equivalent Sound Level in acoustics.

Key Formulas

Speed of Sound

$c = \sqrt{\gamma R T}$

Speed of sound in an ideal gas.

Sound Level

$L = 10 \log_{10}(I/I_0)$

Decibel level.

Wavelength

$\lambda = c / f$

Wavelength = speed / frequency.

Variables

EPA Sound Descriptors

The Environmental Protection Agency (EPA) has adopted a system of four sound descriptors to summarize how sound is heard and measured, and to determine the impact of noise on health and welfare:

Worked Example

A machine generates a sound pressure of 0.02 Pa half the time when idling and 0.1 Pa the other half when working.

Step 1: Calculate $L_{eq}$ for idling (30 min)

$L_{eq,\text{idling}} = 10 \log\left(\frac{1}{0.5\ \text{h}} \left(\frac{0.02\ \text{Pa}}{2 \times 10^{-5}\ \text{Pa}}\right)^2 \times 0.5\ \text{h}\right) = 60\ \text{dB}$

Step 2: Calculate $L_{eq}$ for working (30 min)

$L_{eq,\text{working}} = 10 \log\left(\frac{1}{0.5\ \text{h}} \left(\frac{0.1\ \text{Pa}}{2 \times 10^{-5}\ \text{Pa}}\right)^2 \times 0.5\ \text{h}\right) = 74\ \text{dB}$

Step 3: Calculate total $L_{eq}$ for 1-hour operation

Using the sound pressure method:

$L_{eq,\text{total}} = 10 \log\left(\frac{1}{1\ \text{h}} \left[\left(\frac{0.02}{2 \times 10^{-5}}\right)^2 \times 0.5 + \left(\frac{0.1}{2 \times 10^{-5}}\right)^2 \times 0.5\right]\right) = 71.2\ \text{dB}$

Or equivalently, combining the individual $L_{eq}$ values:

$L_{eq,\text{total}} = 10 \log\left(\frac{0.5 \times 10^{60/10} + 0.5 \times 10^{74/10}}{1}\right) = 71.2\ \text{dB}$

Additional Notes

• $L_{eq}$ is sometimes referred to as Average Sound Level ($L_{AT}$), although this designation is not formally standardized.
• The equivalent sound level correlates well with the physiological and psychological effects of noise on people, making it a preferred metric for noise impact assessments.

References

• Original Source