A guide to decibels and how to use themArticle by Ian Drumm,

Logarithms and Decibels

In acoustics we find huge changes in measurable physical parameters (e.g. pressure, power and voltage)correspond to relatively small changes in perceived loudness. Simply using a linear scale is impractical since thelarge variation in numbers gives little guide to the perceived sound. Instead it is more practical to convert to alogarithmic scale.

The key to understanding acoustic parameters and problems is learning how to use logarithms. Just about everypiece of audio equipment (microphones, loudspeakers, sound cards, amplifiers, mixers, etc) will have specificationsexpressed logarithmically (i.e. in dBs).

The principle of logarithms is fairly straightforward; they are simply a way of expressing parameters that vary byvery large amounts as parameters that vary by relatively small amounts. For example think of the number onemillion

1000000

we can express this as

10 x 10 x 10 x 10 x 10 x 10

i.e.

106

we call 6 the index and 10 the base.

Similarly 100000 = 105, 10000 = 104, 1000=103, 100=102, 10=101, 0=100; so why not express numbers from 0 to 1000000 as numbers from 0 to 6.

Any positive real number can be expressed as an index to a base, written more generally

You can think of a logarithm the result of a function that gives the index of a number with respect to a base

We will be using log to the base 10, hence

(N.B. log10 is often just written as log)

So a log is just an index to a number and hence can be manipulated according to similar laws. For example

Where two numbers multiply we can add their logs.Where two numbers divide we can subtract their logs.Where a number is raised to a power we can multiply the log of that number by the power.

Writing algebraically

Another important rule is

we can use this rule to set a zero point of our log scale. So for instance if we wanted a log to base 10 scale for to have zero at , we would use

Useful parameters such as pressure, power and voltage can be given as a ratio with respect to some fixed reference (e.g. threshold of hearing, 1 Volt, etc)determining the zero point on the scale.

Decibels

The bel scale (named after inventor of telephone) is a means of expressing power as logs to the base 10. However to create a more meaningful range ofnumbers the common convention is to also multiply by 10 (hence decibels - i.e. a decibel is one tenth of a bel).

Converting back from decibels

2 Power and Sound Power Level

An elastic medium can propagate energy as a wave. It is useful to describe energy passing any given point with respect to time . Power is the rate ofenergy flow, where

in Watts(W).

3 Power Level

Consider things that generate sound (e.g. a loudspeaker, lawn mower, violin, etc), most are driven by a steady stream of energy with respect to time (i.e.power) from say an electrical circuit or physical driver. Note energy cannot be created or destroyed, merely transformed. Hence a proportion of the energyinput will be output as acoustic power. The rest is usually dissipated as heat.

When discussing power a logarithmic scale to the base 10 is often appropriate due to typically large variations. Hence

The above gives a Power Level (in dBs) where is power (in watts) given with respect to a reference power .

Power taken with respect to a reference is often referred to as Absolute Power. Alternatively power taken with respect to input power is referred to as Gain.

When quoting a power in terms of logs use

dB re 1.0W

if the reference power is 1.0W

For example:

Calculate power level for 7W using 1.0W as a reference.

Power Level=10*log(7/1) = 8.5dB re 1.0W

It is often useful to convert back to power from power level using

An important term is Sound Power Level (Lw, PWL or SWL) used to describe the acoustic power from a source. This will typically be a small percentage of the

Power driving the source.

The convention for Sound Power Level is to use the reference 10-12W (i.e. 1pW pico watt) corresponding to the minimum acoustic power audible by ahealthy adult.

Hence Sound Power Level is given by

4 Sound Propagation in Free Field

Lets consider a sound propagating from an omni-directional point source without any obstacles. The wave front of propagation is an ever-expanding sphere.As the surface area of this sphere increases with distance the sound energy within a given area decreases. i.e. the intensity of the sound decreases.

The Inverse Square Law - The intensity of sound in a free field is inversely proportional to the square of the distance from the source.

Intensity is the sound energy passing through an area of one meter squared during one second. Hence given the surface area of a sphere is we canexpress the intensity of sound from a point source as

Intensity is difficult to measure, however a related parameter RMS Pressure is easy to measure.

5 RMS Pressure

Air consists of millions of molecules moving around in seemingly haphazard motion. When air is next to a surface it exerts a net force. Pressure is a measure

of this force per unit area .

where the unit of pressure is Pascals (Pa).

Sound modulates this constant atmospheric pressure with its own acoustic pressure (i.e. sound pressure).

Although the sound pressure rapidly varies as peaks and troughs as its compression wave hits a surface, the rate of variation is so high that most instrumentsdetect a wave of overall constant power imparted and hence in effect a constant pressure. This is the Root Mean Squared pressure where

given is the peak pressure of the wave.

You can think of RMS pressure as analogous to the rectified (DC) amplitude of an alternating (AC) signal.

We can directly measure the RMS pressure levels of common sound sources.

Sound Source Pressure (Pa)

Threshold of Pain 112.5

Riveter 20

Speech 0.02

Leaves Rustling 0.0002

Threshold of Hearing 0.00002

As you can see there is a large variation between quiet and loud sounds. This linear measuring scale is hence awkward to use since small variations inloudness are represented by large variations in sound pressure.

6 Sound Pressure Level

A logarithmic scale to the base 10 is more appropriate when describing sound.

Pressure levels in Pascals (Pa) are taken with respect to a reference and converted to logs. The resulting Sound Pressure Levels are hence in decibels(dBs).

where is Sound Pressure Level in dBs relative to , the threshold of hearing, i.e.

0.00002 Pa.

or

Our equation for SPL comes from our equation for sound powerlevel.

The power imparted by sound at a detector is proportional to the square of its pressure, i.e.

hence

Where

7 Some typical SPLs

As you can see the dB scale gives a more sensible description of sound sources.

Sound Source SPL (dB)

It is often useful to be able to convert back from Sound Pressure Levels in dB to pressure levels in Pa using

8 SPLs and Distance

Using logarithms prove useful in dealing with a number of problems where sound pressure levels are involved. For instance we know from the inverse squarelaw that intensity

where is distance from source in metres.

Hence pressure decreases in direct proportion to r, if you have a source at twice its original distance the pressure will be halved.

or using the rule of logs

So more generally for SPLs we can show that

Effectively for every doubling of distance there is 6dB drop in sound pressure level.

9 Combining SPLs

We know that energy and hence power cannot be created or destroyed. Hence given two incoherent sources their combined effect is the sum of their acousticpower.

We can convert SPL to power knowing that

hence

Hence given two sound sources with sound pressure levels and . The combined sound pressure levels are