In a college library a whisper carries about \(10^{-9}\,\text{W/m}^2\) to your ear. On the apron of the airport in Delhi, a jet engine close by carries about \(100\,\text{W/m}^2\). That is \(10^{11}\) times as much, a hundred thousand million. The same ear hears both. This lesson can be read on its own, whenever a physics lesson sends you here.
A range too wide for a ruler
Try to mark both sounds on an ordinary ruler. Let the full length, say \(10\,\text{cm}\), represent the jet's \(100\,\text{W/m}^2\). Then the whisper sits \(10^{-11}\) of the way along, about a hundredth of the width of an atom from zero. Normal talk, at \(10^{-6}\,\text{W/m}^2\), is no better placed. Every everyday sound is crushed against the zero of a linear scale.
A logarithmic scale marks each number at a distance proportional to its logarithm. Then every factor of ten takes the same length, whatever the size of the numbers. The whisper and the jet are eleven factors of ten apart, so they are eleven equal steps apart. The ear works in the same way, as the chapter Energy, Power, Intensity and Loudness explains. It judges loudness by ratios of intensity, not by differences.
The sound level of a sound of intensity \(I\) is \(\beta = (10\,\text{dB})\log_{10}(I/I_0)\). The reference \(I_0 = 1.0 \times 10^{-12}\,\text{W/m}^2\) is the threshold of hearing, so \(0\,\text{dB}\) is the faintest audible sound, not silence. The unit is the decibel, a tenth of a bel.
The whisper has \(I/I_0 = 10^{-9}/10^{-12} = 10^3\), so \(\beta = 10 \times 3 = 30\,\text{dB}\). The jet has \(I/I_0 = 10^{14}\), so \(\beta = 140\,\text{dB}\). Normal talk at \(10^{-6}\,\text{W/m}^2\) is \(60\,\text{dB}\). A range of \(10^{14}\) in intensity has become a range of \(140\) in level.
The whistle of a pressure cooker gives an intensity of \(5.0 \times 10^{-3}\,\text{W/m}^2\) in a small kitchen. What is its sound level, in dB? Take \(\log 5 = 0.70\).
Ratios become differences
Subtract the levels of two sounds. The logarithm of a quotient is the difference of the logarithms, so the reference \(I_0\) cancels: \(\beta_2 - \beta_1 = (10\,\text{dB})\log(I_2/I_1)\). A ratio of intensities becomes a difference of levels. Three benchmarks follow. A factor of \(10\) adds \(10\,\text{dB}\), a factor of \(100\) adds \(20\,\text{dB}\), and a factor of \(2\) adds \(10\log 2 = 3.0\,\text{dB}\).
The figure shows the whole idea of a logarithmic scale. Equal ratios take equal steps, however large or small the numbers are. The same idea sets the musical scale: each octave doubles the frequency, and the ear hears every octave as the same step of pitch.
The intensity of a sound grows by a factor of 100. By how much does the sound level grow?
A rock concert is at \(110\,\text{dB}\) and a quiet conversation at \(50\,\text{dB}\). How many times more intense is the concert?
Sources add intensities, not levels
Two sounds that are not in step add their intensities. Two equal sources give twice the intensity of one. So they add \(3\,\text{dB}\), not twice the level. Ten equal sources give ten times the intensity and add \(10\,\text{dB}\). Adding the levels themselves would give absurd answers. Two sources of \(80\,\text{dB}\) would make \(160\,\text{dB}\), louder than a jet engine.
A sound wave has an intensity of \(2.0 \times 10^{-6}\,\text{W/m}^2\). What is its sound level \(\beta\) in decibels? Then find the level of two identical auto-rickshaws at a signal, if one alone gives \(80\,\text{dB}\).
(a) \(\beta = 10\log(2.0 \times 10^{-6}/10^{-12}) = 10\log(2.0 \times 10^6)\).
The logarithm of a product is the sum of the logarithms: \(\log 2.0 + \log 10^6 = 0.30 + 6 = 6.30\). So \(\beta = 63\,\text{dB}\).
(b) One auto-rickshaw gives \(I_1 = 10^{-12} \times 10^{80/10} = 1.0 \times 10^{-4}\,\text{W/m}^2\). Two give \(2.0 \times 10^{-4}\,\text{W/m}^2\).
Then \(\beta = 10\log(2.0 \times 10^8) = 10 \times 8.30 = 83\,\text{dB}\). Doubling the intensity added \(3\,\text{dB}\).
The phenomenon is answered. The whisper sits at \(30\,\text{dB}\) and the jet at \(140\,\text{dB}\). The step between them, \(110\,\text{dB}\), means eleven factors of ten in intensity. On this scale every step of \(10\,\text{dB}\) means the same thing, a factor of ten, wherever it is taken.
One Diwali cracker gives \(70\,\text{dB}\) at your ear. A string of ten identical crackers bursts together at the same distance. What is the sound level of the string, in dB?
Is a sound at \(70\,\text{dB}\) twice as intense as a sound at \(35\,\text{dB}\)?
A logarithmic scale has no place for zero. A sound of zero intensity has no level, since \(\log 0\) does not exist; \(0\,\text{dB}\) is the threshold of hearing. A negative level is allowed and means a sound fainter than \(I_0\). Also note the factor: the decibel uses \(10\log\) for an intensity or a power. For a pressure amplitude \(p\), whose square is proportional to the intensity, the same level is \(20\log(p/p_0)\).
The decibel scale is the language of the chapter Energy, Power, Intensity and Loudness. It explains why the ear hears ratios, why two sources add \(3\,\text{dB}\), and where the thresholds of hearing and of pain lie. The same scale gives a ladder of common sound levels and the audible range of frequencies, where each octave is one step of pitch. Logarithmic axes also appear whenever physics plots quantities that span many factors of ten.
- Adding the levels of two sources.Add the intensities, then take the level. Two equal sources add only 3 dB.
- Reading a difference of levels as a ratio.A difference \(\Delta\beta\) means a ratio of \(10^{\Delta\beta/10}\): 60 dB is a factor of a million.
- Taking the logarithm of the intensity itself, with its unit.Divide by \(I_0\) first. Only the pure number \(I/I_0\) has a logarithm.