The Logarithmic Scale of Sound
The human ear is an incredible instrument. It can hear the drop of a pin, but it can also withstand the roar of a jet engine without instantly failing. The jet engine produces roughly one trillion times more sound energy than the pin drop.
Because dealing with numbers ranging from $0.000000000001$ to $100$ is mathematically frustrating, scientists use the Decibel (dB) scale. It is a logarithmic scale that compresses this massive range of intensities into manageable numbers from $0$ to $140$.
Understanding the Logarithmic Scale
- $0 , \text{dB}$: The threshold of hearing ($I_0 = 10^{-12} , \text{W/m}^2$). This does not mean "no sound", it means the lowest sound humans can hear.
- $+10 , \text{dB}$: Every increase of $10 , \text{dB}$ means the physical sound intensity has multiplied by $10$.
- $+3 , \text{dB}$: Every increase of $3 , \text{dB}$ means the physical sound intensity has doubled.
The Formula
Where:
L=
Sound Level (Decibels, dB)I=
Sound Intensity (W/m²)=
Reference Intensity (10⁻¹² W/m²)Example Calculation
A rock concert produces a sound intensity of $0.1 , \text{W/m}^2$.
- Divide by Threshold ($I_0$): $0.1 / 10^{-12} = 100,000,000,000$ ($10^{11}$).
- Take Log10: $\log_{10}(10^{11}) = 11$.
- Multiply by 10: $10 \times 11 = 110 , \text{dB}$.
The concert is operating at $110 , \text{dB}$.