Loudness:

i. Loudness depends upon the intensity of vibration.

ii. Intensity of a wave is proportional to square of the amplitude (I ∝ A²) and is measured in the (SI) unit of W/m²

iii. The human response to intensity is not linear, i.e., a sound of double intensity is louder but not doubly loud.

iv. Under ideal conditions, for a perfectly healthy human ear, the least audible intensity is I₀ = 10^-12 W/m².

v. Loudness of a sound of intensity I

(measured in unit bel) is given by,

L₂ = log₁₀ \((\frac{I}{I_0})\) ………….. (1)

vi. Decibel is the commonly used unit for loudness.

vii. As, 1 decibel or 1 dB = 0.1 bel.

∴ 1 bel = 10 dB. 

Thus, loudness in dB is 10 times loudness in bel.

∴ L_dB = 10L_bel = 10 log₁₀ \((\frac{I}{I_0})\)

For sound of least audible intensity I₀

L_dB = 10 log₁₀ \((\frac{I_0}{I_0})\) = 10 log₁₀ (1) = 0 ………… (2)

This corresponds to threshold of hearing.

viii. For sound of 10 dB,

10 = 10 log₁₀ \((\frac{I}{I_0})\)

∴ \((\frac{I}{I_0})\) = 101 or I = 10 I0

For sound of 20 dB,

20 = 10 log₁₀ \((\frac{I}{I_0})\)

= \((\frac{I}{I_0})\) = 10² or I = 100 I₀ and so on.

ix. This implies, loudness of 20 dB sound is felt double that of 10 dB, but its intensity is 10 times that of the 10 dB sound. Similarly, 40 dB sound is left twice as loud as 20 dB sound but its intensity is 100 times as that of 20 dB sound and 10000 times that of 10 dB sound. This is the power of logarithmic or exponential scale. If we move away from a (practically) point source, the intensity of its sound varies inversely with square of the distance, i.e., I ∝ \(\frac{1}{r^2}.\)