Transmission of Sound in Open Space
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2. Transmission of Sound in Open Space

2.1 Point Source

Let us consider a point source of acoustic power W watts, suspended in mid-air and radiates sound waves uniformly in all directions. The sound will spread out spherically and the intensity at any location will depend on the surface area of the sphere at that location (see Figure l). The sound intensity, I, in Watt/m2, at any distance r meters from the source is given by:

(1)

or,

(2)

The above two are alternative expressions of the Inverse Square Law which state that the sound intensity (outward) is inversely proportional to the square of the distance from the point source.

Figure 1 Inverse Square Law of Soun Wave

Recalling the definition of decibels, we have,

and

Let us compare the sound intensity and pressure levels at two distances, r1 and r2 from the sound source. We have,

(3)

If r2 = 2 r1, then

(4)

 

In the case of industrial noise sources, the same general characteristics apply except that usually the problem is much more complicated, owing to the multiplicity of noise sources within a single machine as well as the proximity of other noisy machinery. Nonetheless, it is absolutely necessary to estimate at least the directional characteristics of the noise source in question. This requirement is met by an estimation of the directivity factor, Q, directly or by an approximate calculation using the directivity index, DI.

Figure 2 Sound-Pressure-Level Polar Plot of the Radiation Pattern of a Loudspeaker

The directivity factor, Q, is defined as the ratio of the intensity (W/m2) at some distance and angle from the source to the intensity at the same distance, if the total power from the source were radiated uniformly in all directions.

(5)

where Iq = Sound intensity at distance r and angle q from the source

I = Average sound intensity over a spherical surface at the distance r

And the directivity index (DI) is defined as :

(6)

To take account of any directional effect of the sound source, a directivity factor Q is defined such that the sound intensity will be given by:

(7)

Using equation (7) it can be proved that :

(8)

where,

Lp = sound pressure level, dB

Lw = sound power level of source, dB

r = distance from the source

Q = directivity factor

The values of Q for some ideal geometry and assuming perfect reflection are given in the Table 1.

 

 

Geometry

Value of Q

No surface near sound source; able to radiate acoustical energy in all directions

1

Sound source close to a flat surface; able to radiate acoustical energy to half of a sphere

2

Close to two adjacent flat surfaces perpendicular to each other; able to radiate to one fourth of a sphere

4

At a corner; able to radiate acoustical energy to one eighth of a sphere

8

Table 1 Values of Directivity Factor for Some Ideal Geometry

 

Sphere01.jpg (31839 bytes)

 

Sphere02501.jpg (21898 bytes)

 

If the more common situation applied where the sound is radiated over non-absorbent ground i.e. Q = 2, the equation (6) would become:

(9)

 

2.2 Line source

A line source could be considered as being made up from a line of point sources for each of which the inverse square law applies. It is easier to consider the line radiating in cylindrical form (see Figure 3). The surface area of a cylinder is proportional to its radius. Sound intensity will therefore decrease directly with distance from a line source.

Figure 3 Cylindrical Waves Radiating Out from a Line Source

 

Cylind01.jpg (52216 bytes)

Since the surface area of a cylinder of radius r and length l is , it can be proved that the ratio of sound intensity level for two perpendicular distances R & r from the line is given by :

(10)

If R = 2r, then

It can be seen that there is only 3 dB reduction for a doubling of the distance.

 

2.3 Plane source

For a plane sound source, the sound intensity level and the sound pressure level will both be independent of the distance from the plane assuming that there is no loss in acoustic energy. The sound wave generated by a train or vehicle in a tunnel and propagating towards the tunnel openings is an example of plane source.