Sound Intensity and Loudness
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Direct answer
Intensity of a sound wave is the power transported per unit area perpendicular to propagation, I = P/A, which for a point source spreading spherically becomes I = P/(4πr^2) — the inverse square law, so tripling the distance cuts intensity to a ninth. Intensity relates to wave parameters through I = ½ρvω^2s_m^2 (ρ medium density, v wave speed, s_m displacement amplitude), so doubling the amplitude quadruples the intensity. Because the ear responds to intensity over about twelve orders of magnitude, loudness is measured on the decibel scale β = 10 log_10(I/I_0) with the threshold of hearing I_0 = 10^-12 W/m^2: every factor of 10 in intensity adds 10 dB, and every doubling adds about 3 dB.
What you must remember
- Definitions: intensity I = P/A (W/m^2); for a point source, I = P/4πr^2, giving I ∝ 1/r^2 and amplitude ∝ 1/r.
- Decibel formula: β = 10 log(I/I_0) with I_0 = 10^-12 W/m^2; β(threshold of hearing) = 0 dB and β(threshold of pain) ≈ 120 dB (1 W/m^2).
- Arithmetic anchors: ×10 in intensity = +10 dB; ×2 ≈ +3 dB (since log 2 = 0.30); ×100 = +20 dB; 60 dB means 10^-6 W/m^2.
- Intensity from wave parameters: I = ½ρvω^2s_m^2, intensity ∝ (amplitude)^2 ∝ (frequency)^2 — high-pitched notes of equal amplitude carry far more intensity.
- Loudness versus pitch versus timbre: loudness tracks intensity (and amplitude), pitch tracks frequency, quality distinguishes instruments on the same note — the NCERT trio of sensation attributes.
- Reference levels worth quoting: normal conversation about 60 dB, busy traffic about 80 dB, whispers near 10-20 dB; sustained exposure above roughly 85 dB damages hearing (a widely quoted safety figure).
- Standing-distance trick: if r doubles, I falls to I/4 and β drops by 10 log 4 = 6 dB.
A worked decibel calculation
A loudspeaker radiates 60 W of sound uniformly; find the intensity and sound level at 5 m, then at 10 m. At 5 m: I = 60/(4π × 25) = 60/314 ≈ 0.19 W/m^2, so β = 10 log(0.19/10^-12) = 10 log(1.9 × 10^11) ≈ 10 × 11.28 ≈ 113 dB — near the pain threshold. At 10 m, the radius doubles, so intensity quarters to 0.0476 W/m^2, and the level drops by 10 log 4 ≈ 6 dB to about 107 dB. Notice the habit: never recompute logs from scratch — travel through ratios ("doubling distance costs 6 dB") and only touch the log table once. One more NEET-typical inversion: two identical machines each produce 70 dB; together they deliver twice the intensity, which is 73 dB, not 140 dB — decibels add only 3 for a doubling, and "140 dB" is the option printed for those who added levels directly.
Where students slip
Adding decibels as if they were intensities is the number-one error: two 70 dB sources give 73 dB, ten give 80 dB — the logarithm compresses, and options containing 700 dB-style absurdities trap the hurried. The second slip is forgetting the 4π in point-source problems or using r in centimetres; the intensity then misplaces by orders of magnitude, which the dB scale turns into tens of decibels. Students also quote the threshold of hearing as 0 intensity; it is 10^-12 W/m^2, which the scale renders as 0 dB — the number is zero, the intensity is not. Finally, do not blend intensity (objective, W/m^2) with loudness (subjective sensation, phons at 1 kHz); exam statements carefully say "intensity" when they mean the measurable quantity, and assertion questions exploit the imprecision.
Frequently asked questions
What is the inverse square law for sound intensity?
A point source radiating power P gives intensity I = P/(4πr^2), so intensity falls as 1/r^2 with distance and amplitude as 1/r.
How is the decibel level of a sound computed?
β = 10 log_10(I/I_0) with I_0 = 10^-12 W/m^2 (threshold of hearing); each tenfold intensity change shifts the level by 10 dB.
Two sounds differ by 3 dB — by what factor do their intensities differ?
By a factor of about 2, since 3 dB = 10 log(I_2/I_1) implies I_2/I_1 ≈ 2 (10^0.3).
How does intensity depend on amplitude and frequency?
I = ½ρvω^2s_m^2, so intensity scales as amplitude squared and frequency squared for a given medium.
What distinguishes loudness, pitch and quality of a musical note?
Loudness depends on intensity, pitch on frequency, and quality (timbre) on the harmonic content — which is why a sitar and a flute playing the same note sound different.