Sound Intensity & Decibels
Sound energy spreads out equally in every direction from its source, like ripples growing outward on a sphere. As that sphere gets bigger, the same total power is stretched over more and more area, so the intensity — power per square meter — drops off fast the farther you get, following an inverse-square pattern with distance.
The formula
I = P / (4π · r²)
- I — intensity (W/m²): how much sound power passes through each square meter of area
- P — power (W): the total sound energy the source emits every second
- r — distance (m): how far you are from the source
Worked example
A speaker emits 12.56 watts of sound power equally in all directions. Using 4π ≈ 12.56, what is the sound intensity 1 meter away?
- P = 12.56 W
- r = 1 m
- I = P / (4π · r²)
- I = 12.56 W / (12.56 · 1 m²)
I = 1 W/m²
Test yourself
The same 12.56-watt speaker is now measured 2 meters away instead of 1. What is the new intensity?
- Correct answer: 0.25 W/m²
- 0.5 W/m²
- 4 W/m²
Right! Doubling the distance divides intensity by four (2²): 12.56 W ÷ (4π · 4 m²), using 4π ≈ 12.56, gives 0.25 W/m².
A sound's intensity jumps from I₀ to 100 times I₀. How many decibels does the sound level go up?
- 100 dB
- 10 dB
- Correct answer: 20 dB
Exactly — every ×10 in intensity adds 10 dB, and ×100 is two factors of ×10, so the level rises by 20 dB.
Where you see this
Walk backward from a street busker or across a concert lawn and the loudness falls faster than distance grows — the same sound power is being smeared over an ever-larger sphere. A speaker pushing 12.56 watts delivers 1 W/m² one meter away but only 0.25 W/m² two meters out, and a listener four rows back inherits a small fraction of the front row's intensity.
Common mistakes
The reflex error is halving: double the distance and surely half the intensity — in fact the r in I = P / (4π · r²) is squared, so doubling the distance divides intensity by four, and tripling it divides by nine. The decibel scale trips people the same way: it is logarithmic, so each ×10 in intensity adds only 10 dB — a ×100 jump raises the level by 20 dB, not by a factor of 100.
How it connects
This closes the waves unit: the Doppler shift told you what motion does to frequency, beats what interference does to loudness, and the inverse-square law what distance does to energy — all three act on the same spreading wave. The geometry is identical to the inverse-square falloff of gravity and Coulomb's law: anything radiating equally in all directions spreads over a sphere whose area grows as 4π · r².