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3.6 MEASUREMENT OF SOUND POWER

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The sound pressure in the vicinity of a noise source is generally dependent

on the surroundings. The sound pressure level will be different for the same

noise source, for example, if the source is located indoors or if it is located

outdoors. The sound pressure will be different if the source is placed in a

room with acoustically reflective surfaces or if the room surfaces are highly

absorbent for sound. In contrast, the sound power is generally independent

of the surroundings. For this reason, information about the sound power

spectrum for a noise source is important to the designer interested in noise

control.

There is no ‘‘acoustic wattmeter’’ available for direct measurement of

sound power, however. The sound power must be inferred (or calculated)

from measurements of sound pressure or sound intensity and other appropriate

quantities, such as surface area, reverberation time, etc.

Acoustic Measurements 51

FIGURE 3-8 Personal noise dosimeter. Data can be stored in several locations to

allow the monitoring of multiple inputs. (By permission of Casella CEL Instruments

Ltd.)

There are three broad classes of environment used in connection with

sound power determination: (a) reverberant field, (b) direct or anechoic

field, or (c) the actual environment to which the noise source is exposed

(in-situ survey). The national and international standards for sound power

measurement in a reverberant room include ANSI S1.31, ANSI S1.32, and

ANSI S1.33 (Acoustical Society of America, 1986a,b,c) and ISO 3741, ISO

3742, ISO 3743 (International Organization for Standardization, 1986a,b,c).

The corresponding standards for an anechoic room include ANSI S1.35

(Acoustical Society of America, 1979a) and ISO 3745 (International

Organization for Standardization, 1986d). The survey method is covered

by ANSI S1.36 (Acoustical Society of America, 1976b) and ISO 3746

(International Organization for Standardization, 1986e).

A reverberant room is a room in which the acoustic energy from sound

reflected from the room surfaces (reverberant field) is much larger than the

energy transmitted directly from the noise source to a receiver (direct field),

as discussed in Sec. 7.2. All surfaces in a reverberant room are highly reflective

or have a very low surface absorption coefficient. A reverberant room

may be used to determine sound power by either comparison with a calibrated

noise source or by direct measurement of sound pressure.

An anechoic room is a room in which practically all of the acoustic

energy striking the surfaces of the room is absorbed. Because the energy

reflected from the room surfaces is negligible in an anechoic room, the

energy transmitted directly from the source to the receiver is predominant.

Measurements in an anechoic room may be used to determine the directional

characteristics (directivity factor or directivity index) of the noise

source. One modification of the anechoic room is the semi-anechoic room,

in which the floor surface is highly reflective, but the other surfaces in the

room are highly absorptive. An alternative arrangement is to make measurements

outdoors on a reflective surface, such as a parking lot.

In situations where the noise source cannot be moved into a reverberant

or anechoic room, the sound power may be determined from measurements

taken in situ, with appropriate adjustments made for surrounding

surfaces and environment background noise.

3.6.1 Sound Power Measurement in a

Reverberant Room

The determination of the sound power in a reverberant room requires that

the diffuse or reverberant sound field in the room is much larger than the

direct sound field. This requirement results in a practically uniform value of

the acoustic energy density and the acoustic pressure in the room.

52 Chapter 3

The volume of the reverberant room should be such that the wavelength

of the sound waves is much smaller than the dimensions of the room.

The minimum volume for the room should meet the following condition:

V р3_Ю3 ј 9рc=f Ю3 (3-6)

The quantity c is the speed of sound in the air in the room, and f is the

octave band (or 1/3-octave band) center frequency for the lowest-frequency

band considered in the measurements. For air at 300K

(c ј 347:2m=s ј 1139 fps) and a frequency of 125 Hz, the corresponding

wavelength is as follows:

_ ј р347:2=125Ю ј 2:78m р9:11 ftЮ

The minimum reverberant room volume for this condition may be found

from Eq. (3-6).

V р9Юр2:78Ю3 ј 193m3 р6810 ft3Ю

If the room dimensions (height, width, and length) are in the commonly

used ratio 1 : 1:5 : 2:5, the room dimensions must be at least 3.72m (12.2 ft)

high, 5.58m (18.3 ft) wide, and 9.30m (30.5 ft) long. Room dimension ratios

of 1 : 21=3 : 41=3 or 1 : 1:260 : 1:587 have also been used (Broch, 1971). The

volume of the equipment being tested should not exceed 0.01V, where V is

the volume of the room.

The room surfaces in a reverberant room should have surface absorption

coefficients (Sabine absorption coefficient) that are less than about 0.06.

The surface absorption coefficient is discussed in Sec. 7.1.

The sound field in the region near the walls of the room will not be

quite uniform or diffuse, so it is good practice to locate the microphones

such that none are nearer than the smaller value of 1

2 _ or 1m (39 in) from the

walls. For case of a wavelength of 2.78m given in the previous example

2 _ ј 1:36 m), the microphone should be located at least 1m from the walls.

If an array of microphones is used, at least three microphones should be

included in the array. If a single microphone is used, measurements should

be taken at three or more positions around the noise source. The positions

should be spaced at a distance that is the larger of 3р1

2 _Ю or 3 m, where _ is

the wavelength of the lowest-frequency sound to be measured.

The noise source should not be placed at the center of the room,

because many of the resonant modes of the room would not be excited by

the noise source in this position. The noise source is usually placed near the

room wall at a distance not less than the major dimension of the source.

Acoustic Measurements 53

3.6.1.1 Comparison Method

The sound power may be measured by comparison of the measured sound

pressure level in a reverberant room with the sound pressure level of a

reference (calibrated) sound source at the same location. A reference

sound power source was originally designed by a committee of the

American Society of Heating, Refrigeration, and Air Conditioning

Engineers (ASHRAE) in the 1960s (Baade, 1969). Reference sound power

sources are commercially available with calibration accuracies of 0:5dB

for frequencies between 200 and 4000 Hz and 1:0 dB between 100 and

160 Hz and between 5 and 10 kHz.

Several microphones arranged in an array in the room or a single

movable microphone may be used to measure the sound pressure level in

the reverberant field. To ensure that the microphones are in the diffuse field,

the distance between the microphones and the surface of the noise source dm

should meet the following condition (Beranek and Veґ r, 1992, p. 92):

20 log10р1:25dmЮ LW;cal _ Lp;cal (3-7)

The quantity LW;cal is the sound power level of the reference source, and

Lp;cal is the measured sound pressure level produced by the calibrated sound

power source.

The experimental procedure for determining the sound power level for

a noise source, using the comparison method in a reverberant room, is as

follows. First, the energy-averaged sound pressure level in each frequency

band Lp is measured with the test source in operation. Secondly, the test

source is removed and the reference sound source is placed in the same

location, and energy-averaged sound pressure level in each frequency

band Lp;cal is measured with the calibrated reference source in operation.

Thirdly and finally, the sound power level of the test source LW is calculated

from the measured data:

LW ј Lp ю рLW;cal _ Lp;calЮ (3-8)

The values of the sound power level for the calibrated reference source

LW;cal are supplied by the manufacturer of the calibrated source.

If the reverberant field is much larger than the direct sould field, Eq.

(7-17), which relates the sound power level and sound pressure level, may be

written in the following form:

Lp ј LW ю 10 log10р4=RЮ ю 0:1 (3-9)

The quantity R is the room constant, defined by Eq. (7-13). The value of the

room constant remains constant when the test source is replaced by the

calibrated source:

54 Chapter 3

Lp;cal ј LW;cal ю 10 log10р4=RЮ ю 0:1 (3-10)

If we subtract Eq. (3-10) from Eq. (3-9), we obtain the expression given by

Eq. (3-8).

3.6.1.2 Direct Method

If the room constant were known, Eq. (3-9) could be used to determine the

sound power level directly from sound pressure level measurements. If the

acoustic energy density associated with the reverberant sound field is much

larger than that associated with the direct sound field, Eq. (7-14) may be

written as follows:

4 ј

_oc

(3-11)

If the room is highly reverberant, or if the average surface absorption coefficient

__ is small, the room constant from Eq. (7-13) may be written as

follows:

R ј

__So

1 _ __ _ __So (3-12)

The quantity So is the total surface area of the room.

The reverberation time Tr may be used to determine the average surface

absorption coefficient for the room surfaces. The reverberation time,

adjusted for standing wave effects, is given by Eq. (7-34):

Tr ј

55:26V

1 ю

_ _

(3-13)

The quantity V is the volume of the room, c is the speed of sound in the air

in the room, and a is the number of absorption units, given by Eq. (7-30).

The quantity _ ј c=f is the wavelength of the sound at the band center

frequency, and d ј 4V=So is the mean free path for the sound waves in

the room. For small values of the surface absorption coefficient, the number

of absorption units may be approximated by the following, according to Eq.

(7-30),

a ј So ln

1 _ __

_ _

_ __So (3-14)

By comparing Eqs (3-12) and (3-14), we observe that the room constant and

the number of absorption units are approximately equal.

R _ a _ __So (3-15)

Acoustic Measurements 55

If we make the substitutions from Eq. (3-15) into Eq. (3-13) for the

reverberation time, the following result is obtained:

R ј

55:26V

cTr

1 ю

Soc

8Vf

_ _

(3-16)

If we make the substitution for the room constant from Eq. (3-16) into Eq.

(3-11), the following result is obtained for the acoustic power:

W ј

55:26V

4Tr

1 ю

Soc

8Vf

_ _

_oc2 (3-17)

We may convert Eq. (3-17) to ‘‘level’’ form as follows. First, introducing

the reference quantities, we have the following:

Wref ј р p=pref Ю2рV=Vref Ю

рTr=Tref Ю

1 ю

Soc

8Vf

_ _

13:816р pref Ю2Vref

TrefWref_oc2 (3-17a)

where Vref ј 1m3 and Tref ј 1 sec. If we take log base 10 of both sides of

Eq. (3-17a) and multiply both sides by 10, we obtain the final result needed

to determine the sound power level of a noise source from sound pressure

level measurements in a reverberant room:

LW ј Lp ю 10 log10рV=VrefЮ _ 10 log10рTr=TrefЮ ю 10 log10 1 ю

Soc

8Vf

_ _

ю 10 log10

13:816р pref Ю2Vref

TrefWref_cc2

" #

(3-18)

For an ideal gas, the last term in Eq. (3-18) may be written in the

following form:

_oc2 ј

RT р_RTЮ ј _Po ј _Po;ref рPo=Po;ref Ю (3-19)

The quantity Po is atmospheric pressure, _ is the specific heat ratio

р_ ј 1:40 for air), and Po;ref ј 101:325 kPa (14.696 psia). If we substitute

the numerical values for the reference quantities, we obtain:

13:816р pref Ю2Vref

TrefWref_oc2 ј р13:816Юр20 _ 10_6Ю2р1Ю

р1Юр10_12Юр1:40Юр101:325 _ 103ЮрPo=Po;ref Ю ј

0:03896

рPo=Po;ref Ю

10 log10

13:816р pref Ю2Vref

TrefWref_oc2

" #

ј 10 log10

0:03896

рPo=Po;ref Ю

_ _

ј _10 log10рPo=Po;refЮ _ 14:1

56 Chapter 3

With these substitutions, Eq. (3-18) may be written in the following form:

LW ј Lp ю 10 log10рV=VrefЮ _ 10 log10рTr=TrefЮ ю 10 log10 1 ю

Soc

8Vf

_ _

_ 10 log10рPo=Po;refЮ _ 14:1

(3-20)

To ensure that the diffuse or reverberant sound field predominates at

the microphone location, the distance between the microphones and the

surface of the noise source dm should meet the following condition

(Beranek and Veґ r, 1992, p. 93):

dm 3

cTr

_ _1=2

р3-21)

Example 3-2. A reverberant room has dimensions of 6m (19.69 ft) by 10m

(32.81 ft) by 4m (13.12 ft) high. The measured reverberation time for the

room is 3.50 seconds. The air in the room is at 300K (278C or 808F) and

101.3 kPa (14.7 psia), at which condition the speed of sound is 347.2 m/s

(1139 fps). The measured sound pressure level in the 500 Hz octave band due

to the noise from pump in the room is 65 dB. Determine the sound power

level for the pump in the 500 Hz octave band.

The surface area of the room is as follows:

So ј р2Юр10 ю 6Юр4Ю ю р2Юр10Юр6Ю ј 128 ю 120 ј 248m2 р2669 ft2Ю

The volume of the room is:

V ј р10Юр6Юр4Ю ј 240m3 р8476 ft3Ю

The value of the following quantity for a frequency of 500 Hz may be

calculated:

Soc

8Vf ј р248Юр347:2Ю

р8Юр240Юр500Ю ј 0:0897

The sound power level may now be found from Eq. (3-20):

LW ј 65 ю 10 log10р240Ю _ 10 log10р3:50Ю ю 10 log10р1 ю 0:0897Ю

_ 0 _ 14:1

LW ј 65 ю 23:80 _ 5:44 ю 0:37 _ 14:1 ј 69:6dB

The minimum distance of the microphone from the surface of the

pump is given by Eq. (3-21):

Acoustic Measurements 57

dm р3Ю р240Ю

р347:2Юр3:50Ю

_ _1=2

ј 1:333m р4:37 ftЮ

3.6.2 Sound Power Measurement in an Anechoic

or Semi-Anechoic Room

The acoustic power generated by a noise source may also be measured in an

environment such that the direct acoustic field is much larger than the

reverberant field. This situation may be achieved in an anechoic or semianechoic

room or outdoors away from any reflecting surfaces, such as buildings,

walls, etc. The directivity characteristics of the noise source may be

measured in an anechoic room, as discussed in Sec. 3.6.4.

In an anechoic chamber, the room surfaces are treated with acoustic

material such that the surface absorption is practically 100%. The floor of a

semi-anechoic room is highly reflective, but the walls are highly absorptive,

as in the case of an anechoic chamber.

An array of microphones or a single microphone moved to various

positions may be used for making the measurements. The microphone measurement

positions may be located at the same distance a from the center of

the noise source on a spherical or hemispherical surface. The radius a of the

sphere or hemisphere should be at least twice the largest dimension of the

source or four times the height of the source, whichever is larger. The

microphone measurement position should be farther than a distance dm ј 1

4 _ from any room surfaces, where _ is the wavelength corresponding to the

lowest octave band center frequency considered in the measurements.

If the direct acoustic field is the only contributor to the acoustic pressure,

the sound power Wj radiated from the noise source through an area Sj

is related to the acoustic intensity Ij and measured acoustic pressure pj as

follows:

Wj ј IjSj ј

Sjp2j

_oc

(3-22)

The total sound power radiated from the noise source is the sum of the

power radiated through the entire surface around the source:

W ј

Wj (3-23)

The determination of the sound power may be made somewhat more

convenient by dividing the total surface area of the sphere or hemisphere

into Ns equal surface areas.

58 Chapter 3

Sj ј 2_a2=Ns (for a hemisphere) (3-24)

Sj ј 4_a2=Ns (for a sphere) (3-25)

The surface area of the surface bounded by angles _1 and _2, for

example, is illustrated in Fig. 3-9. The area is given by:

S2 ј

р_2

_1 р2_a sin _Юрa d_Ю ј 2_a2рcos _1 _ cos _2Ю (3-26)

Let us consider the case for a hemispherical area or radius a, and use Ns ј 10 equal areas. The top ‘‘cap’’ will be one area рS1Ю, and each of the next

‘‘rings’’ will each be divided into three areas. The surface area of the ‘‘cap’’,

with _o ј 08, is as follows:

S1 ј 2_a2р1 _ cos _1Ю ј р1=10Юр2_a2Ю (3-27)

If we solve for the angle _1, we obtain the following:

cos _1 ј 1 _ 0:100 ј 0:900 and _1 ј 25:848 ј 0:4510 rad

Using Eq. (3-26), we may obtain the following values for the other angles:

cos _2 ј cos _1 _ 0:300 ј 0:600 and _2 ј 53:138 ј 0:9273 rad

cos _3 ј cos _2 _ 0:300 ј 0:300 and _3 ј 72:548 ј 1:2661 rad

cos _4 ј cos _3 _ 0:300 ј 0:000 and _4 ј 90:008 ј 1

2 _ rad

The microphone should be placed at the geometrical centroid of the

surface area segment, as shown in Fig. 3-9. The centroid for the ‘‘cap’’ is

directly at the top of the hemisphere (__1 ј 08Ю. The angle locating the centroid

of the ‘‘band’’ areas may be found from the following expression:

Acoustic Measurements 59

FIGURE 3-9 Coordinates for determining the microphone locations on a measuring

surface of radius a. The small circles denote the points at the centroid of the area

segments.

рcos _1 _cos _2Ю__2 ј

р_2

_ sin_ d_ р3-28)

__2 ј рsin_2 _sin_1Ю _ р_2 cos _2 __1 cos _1Ю

cos _1 _cos _2

(3-29)

If we substitute the numerical values for a 10-microphone system, the following

value is obtained for the centroid of the first ‘‘band’’ area:

__2 ј

sinр53:138Ю_sinр25:848Ю_Ѕр0:9273Ю cosр53:138Ю

_р0:4510Ю cosр25:848Ю_

р0:900_0:600Ю

__2 ј 0:7121 rad ј 40:808

We may repeat the calculations for the other ‘‘band’’ areas to obtain

the following result:

__3 ј 1:1016 rad ј 63:128

__4 ј 1:4196 rad ј 81:348

The vertical distance from the equator (floor for a hemisphere) yj and

the horizontal distance from the vertical axis xj for the microphone locations

may be found, as follows:

y2=a ј cos__ ј 0:757 and x2=a ј sin__2 ј 0:653

y3=a ј cos__3 ј 0:452 and x3=a ј sin__3 ј 0:892

y4=a ј cos__4 ј 0:151 and x4=a ј sin__4 ј 0:989

The specific locations for 10 microphones, according to ISO3744, are shown

in Fig. 3-10.

For the case in which the measuring surface subdivisions are equal in

area, the total sound power may be found from Eqs (3-22) and (3-23), using

the sound pressure measurements:

W ј _IjSj ј

So_p2

_ocNs

(3-30)

The quantity So is the total surface area (So ј 2_a2 for a hemisphere; So ј 4_a2 for a sphere). For a spherical surface, 20 measurements could be used,

for example. The 10 microphone locations below the equator would be at

the same distance below the equator as those above the equator for the

hemispherical surface with 10 microphone locations.

Example 3-3. The sound pressure level measurements given in Table 3-2

were obtained in a semi-anechoic room around a motor. The overall dimen-

60 Chapter 3

Acoustic Measurements 61

FIGURE 3-10 Microphone locations for a hemispherical measurement surface,

using 10 measurement points (open circles). The closed circles denote locations for

an additional 10 microphones for improved accuracy, particularly for sources having

nonsymmetrical sound radiation characteristics.

TABLE 3-2 Data for Example 3-3

Point Elevation, y, m Angle, _ Lp, dB

1 1.250 0.08 86.0

2 0.9375 41.48 81.5

3 0.9375 41.48 82.4

4 0.9375 41.48 81.3

5 0.5625 63.38 70.9

6 0.5625 63.38 72.9

7 0.5625 63.38 68.0

8 0.1875 81.48 79.3

9 0.1875 81.48 78.5

10 0.1875 81.48 80.1

sions of the motor are 500mm(19.69in) long by 300mm(11.81 in) wide and

300mm (11.81in) high. The microphones were located at a distance of

1.250m (49.21 in) from the center of the motor. The air in the room was

at 248C (297.2Kor 75.28F) and 101.6 kPa (14.74 psia), for which the properties

are sonic velocity c ј 345:6m/s (1134 fps); density _o ј 1:191 kg/m3

(0.0744 lbm=ft3), and _oc ј 411:6rayl. Determine the overall sound power

level for the motor.

The surface area for the measurement hemisphere may be calculated:

So ј р2_Юр1:250Ю2 ј 9:817m2 р105:7ft2Ю

The sumof the squares of the acoustic pressure may be calculated in several

different ways. Let us use the following technique:

_p2j

р pref Ю2 ј _10Lpj

=10 ј 1:14439_109

_p2

j ј р20_10_6Ю2р1:14439_109Ю ј 0:45776 Pa2

The acoustic power for the motor may be calculated from Eq. (3-30):

W ј р9:817Юр0:45776Ю

р411:6Юр10Ю ј 1:092_10_3Wј 1:092mW

The sound power level for the motor is as follows:

LW ј 10 log10р1:092_10_3=10_12Ю ј 90:4dB

3.6.3 Sound Power Survey Measurement

There are some situations in which the noise source cannot be moved into a

reverberant room or into an anechoic room. If the noise source is not

located outdoors away from reflective surfaces, both the direct and diffuse

or reverberant acoustic fields will influence the relationship between sound

power and sound pressure. In this case, the microphone array location on a

rectangular parallelepiped shown in Fig. 3-11 may be used to estimate the

sound power from sound pressure measurements. The measured sound

pressure levels must be ‘‘corrected’’ for the presence of any background

noise, as discussed in Sec. 3.7.

The sound power level may be determined by the comparison method

using a calibrated sound power source, as discussed in Sec. 3.6.1.1. Equation

(3-8) may be used to calculate the sound power level LW for the source from

measurements of the sound pressure level Lp with the sound source in

operation, the sound pressure level Lp;cal with the calibrated sound power

source in operation alone, and the sound power level LW;cal given by the

manufacturer of the calibrated source.

62 Chapter 3

The sound power may also be calculated from measurements of the

reverberation time for the space in which the noise source is located. If we

include the acoustic energy directly transmitted through the measurement

area Sm, Eq. (7-14) may be written in the following form:

R ю

_ _

4Sm

R ю 1

_ _

_oc

(3-31)

The quantity pav is the energy-averaged sound pressure from the measurements:

av ј

_p2j

(3-32)

The room constant R may be taken from Eq. (3-16), with the second term in

parenthesis neglected, since the magnitude of this term is usually smaller

than the uncertainty in the sound power determination:

W ј

Smp2

_oc

1 ю

cTrSm

13:816V

_ _

(3-33)

Acoustic Measurements 63

FIGURE 3-11 Microphone locations for rectangular measuring surfaces, using 9

microphone locations. Point 1 is at the center of the top surface, points 2 through

5 are at the corners of the top surface, and points 6 through 9 are at the centers of the

vertical surfaces.

We may convert Eq. (3-33) to ‘‘level’’ form by introducing the reference

quantities, then taking log base 10 of both sides and multiplying through by

10:

LW ј Lp;av ю10 log10рSm=SrefЮ_Kr _10 log10

_ocWref

refSref

_ _

(3-34)

The reference area is Sref ј 1m2, and the quantity Kr is defined by the

following expression:

Kr ј 10log10 1 ю

cTrSm

13:816V

_ _

(3-35)

For air at 258C (298.2K or 778F) and 101.325 kPa (14.696 psia), the characteristic

impedance _oc ј 409:8 rayl. Using this value, we may determine

the numerical value of the last term in Eq. (3-34):

10log10

_ocWref

refSref

_ _

ј 10log10 р409:8Юр10_12Ю

р20_10_6Ю2р1Ю

" #

ј 0:1 dB (3-36)

The sound power level expression may be written as follows, using the value

of 0.1 dB for the last term in Eq. (3-34):

LW ј Lp;av ю10 log10рSm=SrefЮ_Kr _0:1 (3-37)

There may be some situations in which the noise source cannot be

stopped or ‘‘turned off’’ in order that reverberation time measurements

can be made. In these cases, the sound power may be determined, with

some loss in accuracy, by first estimating the roomconstant. Using information

about the room surfaces and the techniques discussed in Chapter 7, the

room constant may be estimated from Eq. (7-13):

R ј

__S0

1___

(3-38)

The quantity __ is the average surface absorption coefficient for the room

surfaces, and S0 is the total surface area of the room in which the noise

source is located. The factor Kr in Eq. (3-34) is given by the following

expression, using the estimate for the average surface absorption coefficient:

Kr ј 10log10 1 ю

4р1___ЮSm

__S0

_ _

(3-39)

The rectangular parallelepiped measuring surface and the key measuring

points for the survey method of determining the sound power are illustrated

in Fig. 3-11 (ISO, 1986e). The reference surface, with dimensions

‘1; ‘2, and height ‘3, is the smallest parallelepiped that can enclose the noise

64 Chapter 3

source. The measuring surface, on which the microphone measurement

points are located, has dimensions of р‘1 ю2dЮ; р‘2 ю2dЮ; and height

р‘3 юdЮ. The dimension d is somewhat flexible; however, a distance of

d ј 1m (39.4 in) is often used for cases in which the largest dimension

‘max of the reference surface is 250mm (9.8 in) or larger. For cases in

which the largest dimension of the reference surface is less than 250 mm,

the dimension d may be taken as any distance from р4‘maxЮ to 1m, but not

smaller than 250mm. For example, if the largest dimension of the reference

surface is 150mm (5.91in), the dimension d could be chosen as any value

from0.60m(23.6 in) to 1.00m(39.4in). However, if the largest dimension of

the reference surface is 50mm(1.97in), the dimension d would be chosen as

250mm (9.8 in) and not р4Юр50Ю ј 200mm, for example.

There are nine key microphone locations, including locations at the

height h ј 1

2 р‘3 юdЮ in the middle of the four vertical faces, at the center of

the top surface, and at each of the four corners of the top surface. Eight

additional microphone locations, including locations at the center of each of

the four edges of the top surface, and locations at the center of each of the

four vertical edges, may be used for additional accuracy.

Example 3-4. A small air compressor has envelope dimensions of 600mm

(23.6 in) wide by 800mm (31.5 in) long by 600mm (23.6 in) high. The compressor

is located in a room having dimensions of 15m (49.21 ft) by 18m

(59.06 ft) by 3.75m(12.30ft) high. The estimated average surface absorption

coefficient for the room is __ ј 0:15. The measurement surface is selected

with a spacing d of 1.00m (3.28ft or 39.4 in) from the compressor envelope

surfaces, such that the dimensions of the measurement surface are 2.60m

(8.53 ft) by 2.80m (9.19ft) by 1.60m (5.25ft) high. The measured sound

pressure level values are given in Table 3-3 for the 500Hz octave band.

Determine the sound power level of the compressor for the 500 Hz octave

band.

The energy-averaged sound pressure level may be determined from the

nine data points and Eq. (3-32):

рpav=pref Ю2 ј р108:20 ю 108:12 ю_ _ _ю107:81Ю=р9Ю ј 1:0972 _ 108

pav ј р20 _ 10_6Юр1:0972 _ 108Ю1=2 ј 0:2095 Pa

Lp;av ј 10 log10р1:0972 _ 108Ю ј 80:4dB

The measurement surface area is as follows:

Sm ј р2Юр2:60 ю 2:80Юр1:60Ю ю р2:60Юр2:80Ю

Sm ј 17:28 ю 7:28 ј 24:56m2 р264:4 ft2Ю

Acoustic Measurements 65

The surface area of the roommay be determined. We may either neglect the

effect of the area covered by the compressor (less than 1% of the room

surface area) or we may include the surface acoustic absorption of the

compressor and exclude the floor area covered by the compressor. Let us

use the first approach, since the floor area covered by the compressor is

small:

So ј р2Юр15ю18Юр3:75Ююр2Юр15Юр18Ю

So ј 247:5ю540:0 ј 787:5m2 р8477 ft2Ю

The factor Kr may be calculated from Eq. (3-39) for this problem:

Kr ј 10 log10 1 ю р4Юр1_0:15Юр24:56Ю

р0:15Юр787:5Ю

_ _

ј 10 log10р1ю0:7069Ю

ј 2:32dB

The sound power level for the 500Hz octave band may be determined

from Eq. (3.37):

LW ј 80:4ю10log10р24:56Ю_2:32_0:1

LW ј 80:4ю13:90_2:33 ј 92:0 dB

3.6.4 Measurement of the Directivity Factor

The directivity factor Q or the directivity index DI, defined by Eqs (2-27)

and (2-28), may be determined frommeasurements of the sound power in an

anechoic or semi-anechoic room. The directivity may also be measured outdoors

far away from reflecting surfaces. If the measurement is made out-

66 Chapter 3

TABLE 3-3 Data for Example 3-4.

The measurement locations are

illustrated in Fig. 3-11

Point Location Lp, dB

1 Top, center 82.0

2 Top corner, front 81.2

3 Top corner, front 82.6

4 Top corner, back 79.6

5 Top corner, back 80.1

6 Vertical side, center 76.7

7 Vertical side, center 79.8

8 Vertical side, center 80.6

9 Vertical side, center 78.1

doors, the microphone should be located at a distance such that the sound

pressure level decreases by 6 dB for each doubling of the distance from

source. Generally, the number of measurement points required for effective

determination of the directivity is larger than that needed for sound power

determination. The measurement ‘‘mesh’’ should be made finer in the

regions where the sound pressure varies rapidly with position.

The directivity factor Q is defined mathematically by Eq. (2-27). The

directivity factor is the ratio of the sound intensity in a specified direction to

the sound intensity for a spherical source having the same overall sound

power:

Q ј

рW=4_r2Ю ј

4_r2I

W р3-40)

The quantity r is the radial distance from the center of the source to the

point at which the intensity I is determined. The directivity index is the

directional characteristics expressed in ‘‘level’’ form, and is defined by Eq.

(2-28):

DI ј 10 log10рQЮ (3-41)

The acoustic power for the source must be measured first before the

directivity factor can be determined. Using measurements of the acoustic

pressure in an anechoic or semi-anechoic room on a spherical or hemispherical

surface of radius a, the directivity factor may be calculated from Eq. (3-

40), using Eqs (3-22) and (3-23) for the sound power:

Q ј

4_a2p2j

_oc_SjIj ј

4_a2p2j

_Sjp2j

(3-42)

If the total surface area is divided into Ns equal areas, Eq. (3-42) may be

simplified by making the substitutions from Eq. (3-24) or (3-25):

Q ј

2Nsp2j

_p2j

2p2j

(sound source on a reflective surface) (3-43)

Q ј

Nsp2j

_p2j

p2j

(sound source suspended freely) (3-44)

The quantity pav is the energy-averaged sound pressure obtained from the

sound power measurements:

av ј

_p2j

(3-45)

Acoustic Measurements 67

The directivity index may be found from Eq. (3-43) or (3-44) by introducing

the reference sound pressure рpref ј 20 mPa), taking log base 10 of

both sides of the equations, and multiplying through by 10:

DI ј Lpj _Lp;av ю3 dB (sound source on a reflective surface)

(3-46)

DI ј Lpj _Lp;av (sound source suspended freely) (3-47)

Example 3-5. Determine the directivity factor and directivity index for the

sound source given in Example 3-3 for an angle of _ ј 08 with the vertical

and for _ ј 41:48.

The square of the energy-averaged sound pressure is:

av ј р_p2

j Ю=Ns ј р0:45776Ю=р10Ю ј 0:045776 Pa2

The measured acoustic pressure at _ ј 08 may be found from the data in

Table 3-2.

p1 ј р20 _ 10_6Юр1086=20Ю ј 0:3991 Pa

The measurements were taken in a semi-anechoic room, so Eq. (3-43)

may be used to evaluate the directivity factor:

Q ј р2Юр0:3991Ю2

р0:045776Ю ј 6:96 for _ ј 08

The directivity index is found from Eq. (3-41):

DI ј 10 log10р6:96Ю ј 8:43 dB for _ ј 08

We may use an alternative method to determine the directivity index. The

average sound pressure level is as follows:

Lp:av ј 10 log10Ѕр0:045776Ю=р20 _ 10_6Ю2_ ј 80:6dB

The directivity index may be calculated from Eq. (3-46) for measurements

taken in a semi-anechoic room:

DI ј 86:0 _ 80:6 ю 3 ј 8:4dB

From the data given in Table 3-2, we observe that the sound pressure

level at 41.48 does not vary more than about 1 dB with the angle ’. Let us

treat the source as an approximately symmetrical source. The average

acoustic pressure level is found by averaging the data from points 2, 3,

and 4:

68 Chapter 3

Lp2;av ј 10 log10Ѕр1=3Юр108:15 ю 108:24 ю 108:13Ю_ ј 81:8dB

p2;av ј р20 _ 10_6Юр1081:8=20Ю ј 0:2461 Pa

We may use Eq. (3-43) to evaluate the directivity factor for _ ј 41:48:

Q_ ј р2Юр0:2461Ю2

р0:045776Ю ј 2:65 for _ ј 41:48

The directivity index for _ ј 41:48 is as follows:

DI_ ј 10 log10р2:65Ю ј 4:23 dB for _ ј 41:48

The directivity index could also be determined from Eq. (3-46):

Lp2;av ј 20 log10р0:2461=20 _ 10_6Ю ј 81:8dB

DI_ ј 81:8 _ 80:6 ю 3 ј 4:2dB

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