2.7 LEVELS AND THE DECIBEL

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The range of the quantities used in acoustics, such as acoustic pressure,

intensity, power, and energy density, is quite large. For example, the undamaged

human ear can detect sounds having an acoustic pressure as small as

20 mPa, and the ear can withstand sounds for a few minutes having a sound

Basics of Acoustics 27

Copyright © 2003 Marcel Dekker, Inc.

pressure as large as 20 Pa. As a consequence of this wide range of magnitudes,

there was an interest in developing a scale that could represent these

quantities in a more convenient manner. In addition, it was found that the

response of the human ear to sound was more dependent on the ratio of

intensity of two different sounds, instead of the difference in intensity. For

these reasons, a logarithmic scale called the level scale was defined.

The level of any quantity is defined as the logarithm to the base 10 of

the ratio of an energy-like quantity to a standard reference value of the

quantity. The common logarithms (base 10) are used, instead of the natural

or napierian logarithms (base e), because the scale was developed years prior

to the advent of hand-held calculators. Common logarithm tables were

much more convenient to use for widely different quantities than natural

logarithm tables. An energy-like quantity (for example, p2) is used, because

energy is a scalar quantity and an additive quantity. This means that all

levels may be combined in the same manner, if an energy-like quantity is

used.

Although the level is actually a dimensionless quantity, it is given the

unit of bel, in honor of Alexander Graham Bell. It is general practice to use

the decibel (dB), where 1 decibel is equal to 0.1 bel. The history of the

development of the bel unit is described by Huntley (1970). The level is

usually designated by the symbol L, with a subscript to denote the quantity

described by the level. For example, the acoustic power level is designated

by LW. The acoustic power level is defined by:

LW ј 10 log10рW=Wref Ю (2-33)

The factor 10 converts from bels to decibels. The reference acoustic power

рWref Ю is 10_12 watts or 1 pW.

The sound intensity level and sound energy density level are defined in

a similar manner, since both of these quantities (I and D) are proportional

to energy:

LI ј 10 log10рI=IrefЮ р2-34Ю

LD ј 10 log10рD=Dref Ю (2-35)

where the reference quantities are:

Iref ј 10_12W=m2 ј 1pW=m2

Dref ј 10_12 J=m3 ј 1pJ=m3

The reference quantities were not completely arbitarily selected. At a

frequency of 1000 Hz, a person with normal hearing can barely hear a sound

having an acoustic pressure of 20 mPa. For this reason, the reference acoustic

pressure was selected as:

28 Chapter 2

Copyright © 2003 Marcel Dekker, Inc.

pref ј 20 mPa ј 20_10_6 Pa

The characteristic impedance for air at ambient temperature and pressure

is approximately Zo _ 400 rayl. The acoustic intensity corresponding to a

sound pressure of 20 mPa moving through ambient air is approximately

Iref ј р20_10_6Ю2=р400Ю ј 10_12W=m2 or 1pW. The acoustic power

corresponding to the reference intensity and a ‘‘unit’’ area of 1m2 is

Wref ј р10_12Юр1Ю ј 10_12W or 1pW. The reference acoustic energy

(Dref ј 1pJ=m3Ю was somewhat arbitarily selected, because the acoustic

energy density for a plane sound wave in ambient air with the reference

sound pressure level is approximately 0.003 pJ/m3.

We note that the acoustic pressure is not proportional to the energy,

but instead, p2 is proportional to the energy (intensity or energy density).

For this reason, the sound pressure level is defined by:

Lp ј 10 log10рp2=p2

refЮ ј 20log10р p=pref Ю (2-36)

The expressions for the various ‘‘levels’’ and the reference quantities,

according to ISO and ANSI, are given in Table 2-1.

One feature of the use of the decibel notation is that many expressions

involve addition or subtraction, instead of multiplication or division. This

feature was advantageous before the advent of hand-held digital calculators

and digital computers. If we combine Eqs (2-13) and (2-16) for the acoustic

intensity, we obtain:

I ј

QW

4 _r2 ј

p2

_c

(2-37)

Basics of Acoustics 29

TABLE 2-1 Reference Quantities for Acoustic Levels

Quantity Definition, dB Reference

Sound pressure level Lp

ј 20 log10

р p=pref

Ю pref

ј 20mPa

Intensity level L1

ј 10 log10

рI=Iref

Ю Iref

ј 1pW=m2

Power level LW

ј 10log10

рW=Wref

Ю Wref

ј 1 pW

Energy level LE

ј 10log10

рE=Eref

Ю Eref

ј 1 pJ

Energy density level LD

ј 10 log10

рD=Dref

Ю Dref

ј 1pJ=m3

Vibratory acceleration level La

ј 20 log10

рa=aref

Ю aref

ј 10mm=s2

Vibratory velocity level Lv

ј 20 log10

рv=vref

Ю vref

ј 10nm/s

Vibratory displacement level Ld

ј 20 log10

рd=dref

Ю dref

ј 10pm

Vibratory force level LF

ј 20log10

рF=Fref

Ю Fref

ј 1 mN

Frequency level Lfr

ј 10log10

р f =fref

Ю fref

ј 1Hz

Note: The SI prefixes are listed in Appendix A.

Source: From ISO Recommendation No. 1683 and American National Standard

ANSI S1.8 (1989).

Copyright © 2003 Marcel Dekker, Inc.

The acoustic pressure is related to the acoustic power:

p2 ј

WQ_c

4_r2 (2-38)

We may introduce the reference pressure and reference power:

p2

p2

ref ј

WQ_cWref

4_Wrefp2

ref r2 (2-39)

If we take the common logarithm of both sides of Eq. (2-39) and multiply

both sides by 10, we obtain:

Lp ј LW ю DI _ 20 log10рrЮ ю 10 log10

_cWref

4_p2

ref

_ _

(2-40)

The quantity DI is the directivity index, defined by:

DI ј 10 log10рQЮ (2-41)

where Q is the directivity factor. If we express the radial distance r in meters

and take the properties of air at 258C (778F), we may determine the numerical

value for the last term in Eq. (2-40):

10 log10 р4_Юр20 _ 10_6Ю2

р409:8Юр10_12Ю

" #

ј 10 log10р12:266Ю ј 10:9dB

For sound transmitted from a directional source (or spherical source,

with DI ј 0 or Q ј 1) outdoors in air at 258C, the sound pressure level and

the sound power level are related by:

Lp ј LW ю DI _ 20 log10рrЮ _ 10:9 dB (2-42)

Example 2-6. The quantities in Examples 2-2 and 2-3 are as follows:

acoustic pressure, p ј 0:20 Pa

acoustic particle velocity, u ј 0:488 mm/s

acoustic intensity, I ј 97:6 mW=m2

acoustic energy density, D ј 0:282 mJ=m3

Determine the corresponding levels for these quantities.

The sound pressure level is:

Lp ј 20 log10р0:20=20 _ 10_6Ю ј 80:0dB

The velocity level is:

Lv ј 20 log10р0:488 _ 10_3=10 _ 10_9Ю ј 93:8dB

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Copyright © 2003 Marcel Dekker, Inc.

The intensity level is:

LI ј 10 log10р97:6 _ 10_6=10_12Ю ј 79:9dB

We note that, for sound transmitted in air at ambient conditions, the intensity

level and the sound pressure level have approximately equal values. This

is a consequence of the definition of the reference quantities for acoustic

pressure and intensity. For sound transmitted in water, on the other hand,

the sound pressure level and intensity level have quite different values.

The energy density level is:

LD ј 10 log10р0:282 _ 10_6=10_12Ю ј 54:5dB

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