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Sound intensity I is the power per unit area (W/m²) carried by the wave. For a point source radiating uniformly, I follows an inverse-square law: I ∝ 1/r². Because the human ear responds logarithmically, the decibel scale is used: β = 10 log₁₀(I/I₀), where I₀ = 10⁻¹² W/m² is the threshold of hearing. Each 10 dB increase represents a tenfold increase in intensity.
Calculate the dB level for common sounds (conversation ≈ 60 dB, concert ≈ 110 dB) and verify that doubling distance reduces level by about 6 dB.
From wave energy, you know that intensity means power per unit area — the energy the wave delivers per second to each square meter of surface it passes through. For a point source radiating equally in all directions, the wave spreads over a sphere of area 4πr². Since the source's total power P is constant, the intensity at distance r is I = P/(4πr²). Double the distance and intensity drops by a factor of four. Triple it and intensity drops by nine. This inverse-square law explains why sounds fade so quickly with distance — standing twice as far from a speaker delivers one-quarter the acoustic power to your ear.
Now connect this to logarithms, which you've studied as a tool for compressing quantities that span enormous ranges. Human hearing is sensitive over an intensity range of roughly 10¹² — from the threshold of hearing at I₀ = 10⁻¹² W/m² to the threshold of pain near 1 W/m². Expressing everyday sounds in watts per square meter produces numbers so small (conversation ≈ 10⁻⁶ W/m²) that comparing them is inconvenient. The logarithm collapses this. The decibel level is defined as β = 10 log₁₀(I/I₀). A whisper at 10⁻¹⁰ W/m² gives β = 10 log₁₀(10⁻¹⁰/10⁻¹²) = 10 log₁₀(100) = 20 dB. A conversation at 10⁻⁶ W/m² gives β = 10 × 6 = 60 dB. A rock concert near 10⁻¹ W/m² gives β = 110 dB. The entire 10¹² range compresses into 0–120 dB — a scale that matches how our ears actually perceive relative loudness.
The most important arithmetic rule: every 10 dB corresponds to a factor of 10 in intensity. This follows directly from the definition: if I₂ = 10 I₁, then β₂ − β₁ = 10 log₁₀(10) = 10 dB. A 70 dB vacuum cleaner is ten times more intense than a 60 dB conversation; a 110 dB concert is 10,000 times more intense than conversation. Two useful derived results: doubling intensity adds 10 log₁₀(2) ≈ 3 dB, and doubling distance (reducing intensity by 4) drops the level by about 6 dB. These approximations are worth memorizing for quick estimates.
A subtle but important point: the decibel scale measures physical intensity, not perceived loudness. The ear's sensitivity varies with frequency — a 1,000 Hz tone at 60 dB sounds much louder than a 100 Hz tone at 60 dB. Audio engineers use loudness-weighted scales (A-weighting, phon curves) that account for this, which is why you'll see "dBA" on noise regulations. For physics problems, however, stick with the standard intensity-based definition. Also remember that decibels are always a ratio relative to a reference level: I₀ = 10⁻¹² W/m² is the standard acoustic reference, but other fields use different references (dBm in radio engineering uses 1 milliwatt). The formula is always β = 10 log₁₀(I/I_reference), and the reference must be stated for the number to have meaning.