2.8 COMBINATION OF SOUND SOURCES

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There are many situations in which we need to determine the sound level

produced by several sources of sound acting at the same time. For example,

we may need to determine the sound level produced by two machines in a

room, but we may have only information about the sound level produced by

each machine separately.

Because all levels are defined in terms of energy-like quantities, all

‘‘levels’’ (sound pressure level, intensity level, power level, etc.) will combine

in the same manner. The total intensity, for example, is the sum of the

intensities for the individual sources, if the sources produce sound waves

that are not exactly in-phase or out-of-phase. It is quite likely that the noise

generated by machinery is not correlated, because the noise involves a wide

range of frequencies and not a single frequency only.

Suppose we have two sources of sound that produce the following

intensity levels when operating alone:

LI1 ј 80 dB ј 10 log10рI1=Iref Ю

LI2 ј 85 dB ј 10 log10рI2=Iref Ю

We may solve for the individual intensities to obtain these values:

I1 ј р10_12Ю10р80=10Ю ј 10_4W=m2 ј 0:100mW=m2

I2 ј р10_12Ю10р85=10Ю ј 3:16 _ 10_4W=m2 ј 0:316mW=m2

The total intensity when the two sources are operating at the same time is

the sum of the intensities:

I ј I1 ю I2 ј 0:100 ю 0:316 ј 0:416mW=m2

Basics of Acoustics 31

Copyright © 2003 Marcel Dekker, Inc.

The combined intensity level is:

L1 ј 10log10р0:416_10_3=10_12Ю ј 86:2 dB

The general expression for determining the combination of any set of

‘‘level’’ quantities is:

L ј 10log10

X

i

10Li=10

!

(2-43)

This expression is valid for all types of levels—including sound pressure

levels—because the total pressure is not the sum of the individual pressures

if the waves are not correlated. The square of the pressure is proportional to

energy (the intensity, for example), so the individual sound pressures must

be combined in an ‘‘energy-like’’ manner.

р ptotalЮ2 ј p2

1 юp2

2 юp2

3 ю_ _ _

The reference quantity is the same for each intensity level, so the

previous calculation could be carried out using the intensity ratio. For the

values used in the previous example, we have:

I1=Iref ј 10р80=10Ю ј 108

The total intensity ratio is the sum of the individual ratios:

I=Iref ј рI1=IrefЮ ю рI2=IrefЮ ј 108 ю3:16_108 ј 4:16_108

The total intensity level is:

LI ј 10log10р4:16_108Ю ј 86:2dB

The previous calculation provides the basis for a simple routine for

combining levels on the hand-held calculator. The routine is outlined in

Table 2-2. Different calculators have slightly different designations on the

keys; however, the routine may be adapted to any calculator. The initial

level value divided by 10 is first entered. (One may easily divide a number by

10 without the use of a calculator.) The 10x key (or the corresponding key to

raise 10 to the power of the entry) is pressed. The result is then stored in

memory. Each of the next levels divded by 10 is entered, the 10x key is

pressed, and the results are added to memory. To finally determine the

total level, the quantity is recalled from memory, and the log key (or logbase-

10 key) is pressed. The display may be multiplied by 10 to obtain the

total level.

32 Chapter 2

Copyright © 2003 Marcel Dekker, Inc.

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