2.6 DIRECTIVITY FACTOR AND DIRECTIVITY INDEX

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The acoustic energy is radiated uniformly in all directions for a spherical

wave; however, other sources of sound may be highly directional. These

directional sources radiate sound with different intensities in different directions.

The intensity of noise radiated from a vent pipe along the axis of the

vent pipe, for example, is different from the intensity perpendicular to the

vent pipe axis. In fact, if a spherical source is placed near the floor or a wall,

some sound will be reflected from the surface and will not be radiated in all

directions.

The directivity factor (Q) is defined as the ratio of the intensity on a

designated axis of a sound radiator at a specific distance from the source to

the intensity that would be produced at the same location by a spherical

source radiating the same total acoustic energy:

Q ј

4_r2I

W

(2-27)

24 Chapter 2

Copyright © 2003 Marcel Dekker, Inc.

The directivity index (DI) is related to the directivity factor by:

DI ј 10 log10рQЮ (2-28)

For a spherical source, the directivity factor Q ј 1 and the directivity index

DI ј 0.

The directivity factor may be determined from analytical or experimental

values of the acoustic pressure. The directional pressure distribution

function Hр_; ’Ю is defined by:

Hр_; ’Ю ј

pр_; ’Ю

pр0Ю

(2-29)

The quantity _ is the azimuth angle, and ’ is the polar angle, as shown in

Fig. 2-4: pр0Ю is the acoustic pressure on the axis, _ ј 0. The directivity

factor may be evaluated from the directional pressure distribution function:

Q ј

4_

р_

0

р2_

0

H2р_; ’Ю sin _ d’ d_

(2-30)

If the pressure distribution is symmetrical, or Hр_; ’Ю ј Hр_Ю, the integration

with respect to ’ may be carried out directly. The directivity factor for a

symmetrical source of sound is given by:

Q ј

2 р_

0

H2р_Ю sin _ d_

(2-31)

Basics of Acoustics 25

FIGURE 2-4 Spherical coordinates for directivity factor.

Copyright © 2003 Marcel Dekker, Inc.

The directivity factor for locations off the axis рQ_Ю may be expressed, as

follows:

Q_р_; ’Ю ј QH2р_; ’Ю р2-32)

If a spherical source of sound is placed near the floor or a wall, as

shown in Fig. 2-5, sound is radiated through a hemispherical area, S ј 2_r2.

In this case, the intensity is:

I ј

W

2_r2 ј

2W

4_r2 ј

QW

4_r2

For this case, we see that the directivity factor is Q ј 2, and the directivity

index is:

DI ј 10 log10р2Ю ј 3:0

26 Chapter 2

FIGURE 2-5 Sound sources near a surface for (A) directivity factor Q ј 2 and

directivity index DI ј 3 and (B) for Q ј 4 and DI ј 6.

Copyright © 2003 Marcel Dekker, Inc.

Similarly, if the spherical source were placed on the floor near a wall, the

energy is radiated through an area S ј _r2. For this case,

I ј

W

_r2 ј

4W

4_r2 ј

QW

4_r2

The directivity factor, in this case, is Q ј 4 and the directivity index is 6.0.

By going through the same reasoning, we may show that if the spherical

source were placed in a corner near the floor and two walls, Q ј 8 and

DI ј 9:0.

From a practical standpoint, these results show the importance of

location of a noisy piece of machinery. If the machine is located on the

floor, it will produce an intensity that is about twice that produced by the

same machine away from the floor. The intensity for the machine located on

the floor near a wall will be about four times that measured with the

machine away from reflective surfaces.

Example 2-5. A source of sound radiates symmetrically with the following

directional pressure distribution function:

Hр_Ю ј cos _

Determine the directivity factor and directivity index in the direction _ ј 0.

The integral in the denominator of Eq. (2-31) may be evaluated first:

р_

0

H2р_Ю sin _ d_ ј

р_

0

cos2 _ sin _ d_ј _1

3 cos3 _

_ __

0ј 2=3

The directivity factor is evaluated from Eq. (2-31).

Q ј

2

2=3 ј 3

The directivity index is found from Eq. (2-28)

DI ј 10 log10р3Ю ј 4:8

The directivity factor at an angle of _ ј 458 from the axis is:

Q_ ј QH2р_Ю ј р3:00Ю cos2р458Ю ј 1:500

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