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Energy, Power, Intensity and Loudness · Lesson 4 of 5

The Decibel Scale

14 min read3 ideas5 checks1 figure to playBy Asheshanand Acharya
What you will learn 0 / 3
The ear hears ratios of intensity
Maths you need
Look around
Ten crackers are not ten times as loud

On Diwali night one cracker bursts in the street, and then a string of ten crackers bursts together. The string delivers about ten times the intensity to your ears. Yet it does not sound ten times as loud. It sounds perhaps twice as loud. Your ear is not measuring intensity the way a meter does.

you will find the scale that matches the way the ear judges loudness, and you will learn to read a sound level in decibels.

The ear hears ratios

Our ears perceive loudness logarithmically, not linearly. A sound 100 times more intense sounds louder by only a few times. What the ear notices is the ratio of two intensities, not their difference. Going from 1 to 10 units of intensity feels like the same step as going from 10 to 100.

This is how the ear copes with an enormous range. Lesson 5 shows that we hear intensities from \(10^{-12}\,\text{W/m}^2\) to about \(1\,\text{W/m}^2\), a range of a million million. An ear that responded in proportion to intensity could not span that range. An ear that responds to ratios needs only twelve steps of ten.

The same tone 10 dB and 20 dB louderA graph of the pressure variation delta P of a 1 kilohertz tone against x, drawn to one fixed scale. Three chips set the level: a starting level L, L plus 10 decibels and L plus 20 decibels. Faint dashed lines mark the three amplitudes, in the ratio 1 to 3.16 to 10, since the pressure amplitude grows as the square root of the intensity: each 10 decibels multiplies the intensity by 10 and the pressure amplitude by 3.16. A readout names the step. A Listen chip plays the tone at the chosen level for three seconds; nothing plays until it is clicked.OxΔP level L: the starting sound
Fig 4.1The same \(1\,\text{kHz}\) tone at a starting level \(L\), then \(10\,\text{dB}\) and \(20\,\text{dB}\) louder. Each step multiplies the intensity by \(10\) and the pressure amplitude by \(\sqrt{10} = 3.16\). Press Listen, then change the chip while the tone plays. Your own speakers set the starting level.

Listen to Fig 4.1 and change the chip. The pressure wave in the figure grows by \(3.16\) at each step, and the intensity by \(10\). Yet each step sounds only about twice as loud as the one before. The step of \(20\,\text{dB}\) multiplies the intensity by a hundred and sounds only about four times as loud.

Story · Gustav Fechner, 1860
Sensation grows as a logarithm

Gustav Fechner, a German physicist, studied how strongly people feel a stimulus. Earlier, Ernst Weber had found that a weight must grow by a fixed fraction, not a fixed amount, before a person notices the change. In his Elements of Psychophysics of 1860, Fechner turned this into a law. The strength of a sensation grows as the logarithm of the stimulus. The decibel scale of sound is built on the same idea.

Sound level in decibels
Maths you need

The decibel scale

To model the ear, we use the decibel (dB) scale. It measures the sound level \(\beta\) of a sound by comparing its intensity \(I\) with a fixed reference intensity \(I_0\). The reference is \(I_0 = 1.0 \times 10^{-12}\,\text{W/m}^2\), the threshold of hearing of a healthy young ear.

definition 6.2
Sound level

The sound level \(\beta\) of a sound of intensity \(I\) is \(\beta = (10\,\text{dB})\log_{10}(I/I_0)\), where \(I_0 = 1.0 \times 10^{-12}\,\text{W/m}^2\) is the threshold of human hearing. The unit is the decibel, dB.

RememberSound level in decibels
\[ \beta = (10\,\text{dB})\log_{10}\left(\dfrac{I}{I_0}\right), \qquad I = I_0 \times 10^{\beta/(10\,\text{dB})} \]
Holds whenany sound; \(I_0 = 1.0 \times 10^{-12}\,\text{W/m}^2\) is the agreed reference intensity.
Symbols\(\beta\) sound level, \(I\) intensity, \(I_0\) reference intensity, the threshold of hearing
Units\(\beta\) in dB, \(I\) and \(I_0\) in W/m²

The logarithm is a ratio in disguise. At the threshold of hearing \(I = I_0\), so \(\beta = 10\log 1 = 0\,\text{dB}\). A level of \(0\,\text{dB}\) is not silence. It is the faintest sound the ear can hear. The name comes from the bel, named after Alexander Graham Bell, the inventor of the telephone. A decibel is a tenth of a bel.

Intensity on a logarithmic scale and the sound levelA slider sets the intensity I of a sound as a power of ten, from 10 to the minus 12 up to 10 squared watts per square metre. Left, a column on a logarithmic scale: its ticks are every 20 decibels from 0 to 140, with the intensities 10 to the minus 12, 10 to the minus 10 and so on up to 1 and 100 watts per square metre beside them; the shaded height follows the slider, so equal steps of the slider are equal steps of the column. Right, the same intensity on a linear scale from 0 to 100 watts per square metre: the shading stays invisible until the level passes about 120 decibels. Readouts give I and the sound level beta = 10 log of I over I nought; 2.0 times 10 to the minus 6 watts per square metre is 63 decibels. log scale linear scale β (dB) I (W/m²) I (W/m²) 0 10⁻¹² 20 10⁻¹⁰ 40 10⁻⁸ 60 10⁻⁶ 80 10⁻⁴ 100 10⁻² 120 1 140 10² 0 50 100 I = 2.0 × 10⁻⁶ W/m² β = 10 log(I/I0) = 63 dB
SimSet the intensity with the slider, as a power of ten. The left column is a logarithmic scale: each step of \(20\,\text{dB}\) multiplies the intensity by \(100\). The right column shows the same intensity on an ordinary linear scale up to \(100\,\text{W/m}^2\). The readouts give \(I\) and the sound level \(\beta\).

Move the slider in the figure above. On the left column, every step of one power of ten raises the level by the same \(10\,\text{dB}\). On the linear column nothing shows until the level passes about \(120\,\text{dB}\). At \(120\,\text{dB}\) the intensity is \(1\,\text{W/m}^2\), only one hundredth of the column. Every sound we meet in daily life is squeezed into the bottom of a linear scale.

Useful benchmarks

The logarithm turns a ratio of intensities into a difference of levels. So three results are worth remembering.

  • An increase of 10 dB means the intensity has multiplied by 10.
  • An increase of 20 dB means the intensity has multiplied by 100.
  • An increase of 3 dB means the intensity has approximately doubled, since \(10\log_{10}2 = 3.01\).
CheckpointUnanswered

If the intensity of sound increases by a factor of 100, the sound level increases by

CheckpointUnanswered

True or false: a sound at 70 dB has twice the intensity of a sound at 35 dB.

ExampleFrom intensity to sound level

A sound wave has an intensity of \(2.0 \times 10^{-6}\,\text{W/m}^2\). What is its sound level in decibels, \(\beta\)?

Try it on paper first. The solution is one tap away when you are ready.

Set the slider to \(-5.7\) to check the example: the readouts show \(2.0 \times 10^{-6}\,\text{W/m}^2\) and \(63\,\text{dB}\). The intensity of normal conversation in lesson 2, \(1.0 \times 10^{-6}\,\text{W/m}^2\), is exactly \(60\,\text{dB}\). The example's sound is twice as intense, so its level is \(3\,\text{dB}\) higher.

Checkpoint · numericUnanswered

The whistle of a pressure cooker produces an intensity of \(5.0 \times 10^{-3}\,\text{W/m}^2\) in a small kitchen. What is its sound level, in dB?

dB± 2 %
Differences of level are ratios of intensity
Maths you need

Comparing two levels

To compare two sounds, subtract their levels. The reference \(I_0\) cancels, because a difference of logarithms is the logarithm of a ratio. So \(\beta_2 - \beta_1 = (10\,\text{dB})\log_{10}(I_2/I_1)\). Every \(10\,\text{dB}\) of difference is a factor of \(10\) in intensity.

RememberDifference of sound levels
\[ \beta_2 - \beta_1 = (10\,\text{dB})\log_{10}\left(\dfrac{I_2}{I_1}\right), \qquad \dfrac{I_2}{I_1} = 10^{(\beta_2 - \beta_1)/(10\,\text{dB})} \]
Holds whenany two sounds; the reference intensity \(I_0\) cancels.
Symbols\(\beta_1, \beta_2\) sound levels, \(I_1, I_2\) intensities
Units\(\beta\) in dB; the ratio of intensities has no unit
CheckpointUnanswered

A rock concert has a sound level of 110 dB, while a quiet conversation is 50 dB. How many times more intense is the concert than the conversation?

Adding sources

Two sounds that are not in step add their intensities, not their levels. So two identical sources give twice the intensity of one. The level rises by \(10\log_{10}2 = 3\,\text{dB}\), whatever the level of one source. Ten identical sources give ten times the intensity, \(10\,\text{dB}\) more.

ExampleTwo auto-rickshaws at a signal

At a traffic signal, one auto-rickshaw idling beside you gives a sound level of \(80\,\text{dB}\). A second, identical auto-rickshaw stops at the same distance. What is the sound level of the two together?

Reveal one step at a time and finish the rest yourself.
Checkpoint · select all that applyUnanswered

Which of these statements about sound levels are true?

Now return to the Diwali street. The string of ten crackers delivers ten times the intensity of one. On the decibel scale that is only \(10\,\text{dB}\) more. The ear judges a step of \(10\,\text{dB}\) as about twice as loud. So the ten crackers sound roughly twice as loud as one, just as you noticed.

Exam traps
  1. Treating decibels as proportional to intensity.
    The scale is logarithmic. 70 dB is \(10^{3.5} \approx 3162\) times as intense as 35 dB, not twice.
  2. Adding the levels of two sources.
    Add the intensities. Two identical sources add only \(3\,\text{dB}\).
  3. Reading \(0\,\text{dB}\) as silence.
    \(0\,\text{dB}\) is the threshold of hearing, \(I = I_0 = 10^{-12}\,\text{W/m}^2\).
  4. Forgetting to divide by \(I_0\) before taking the logarithm.
    \(\beta = 10\log_{10}(I/I_0)\). The ratio \(I/I_0\) has no unit.