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Appendix I:A-3. Sound Propagation
The sound power level of a source is independent of the environment. However, the sound pressure level at some distance
(r) from the source is dependant on that distance and the sound-absorbing characteristics of the environment. Sound propagation is explained
in terms of:
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Sound Power Level |
Sound Power (W) is the amount of energy per unit time that radiates from a source in the form of an acoustic wave. The
Sound Power Level (Lw), in decibels, is defined as:
Lw = 10 log10 (W/W0)
W = sound power in watts
Wo = reference sound power (10-12 watts)
Doubling the sound power increases the sound power level by 3 decibels (dB).
Example:
W = 2 watts: Lw = 10 log (2/10-12) = 123 dB
W = 4 watts: Lw = 10 log (4/10-12) = 126 dB
In a free field (no reflections), the sound from a point source radiates equally in all directions in the form of a spherical wave. This
distribution of sound power over the area of the propagating wave is designated as intensity and is measured in units of watts per square
meter.
Sound power cannot be measured directly. It is possible to measure intensity, but the instruments are relatively expensive and must be used
carefully. Under most conditions of sound radiation, sound intensity is proportional to the sound pressure. Sound pressure can be measured
more easily, so sound measuring instruments are built to measure the sound pressure level in dB. |
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Sound Pressure Level |
 The Sound Pressure Level (Lp), in decibels, is given by the following:
Lp = 20 log10 (p/p0)
p = measured root-mean-square (rms) sound pressure
p0 = reference rms sound pressure (20 micropascals (µPa))
The reference sound pressure of 20 micropascals approximates the normal threshold of human hearing at 1,000 Hz.
- Note: The multiplier above is 20 and not 10 (as in the case with sound power level). This is because the sound power is proportional to
the square of sound pressure and 10 log p2 = 20 log p.
There is a significant advantage to using decibel notation rather than the wide range of pressure or power.
- Note: A change in sound pressure by a factor of 10 corresponds to a change in sound pressure level of 20 dB:
p = 40 µPa: Lp = 20 log (40/20) = 6 dB
p = 400 µPa: Lp = 20 log (400/20) = 26 dB
Doubling the sound pressure results in an increase of 6 dB in sound pressure level:
p = 40 µPa: Lp = 20 log (40/20) = 6 dB
p = 80 µPa: Lp = 20 log (80/20) = 12 dB
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The Difference Between Sound Power Level and Sound Pressure Level |
There is a common tendency to confuse sound power with sound pressure. Sound power is analogous to the power rating of
a light bulb.
- A "weak" sound source will produce low sound levels, whereas a "stronger" sound source will produce higher sound levels.
- Sound power level is independent of the environmental surroundings.
- The magnitude of the sound pressure from a given sound source, however, depends on the distance from the source and the characteristics
of the environmental surroundings.
- Sound pressure level is readily measured by instruments, but sound power cannot be measured directly.
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Non-Directional Sound in a Free-Field |
The simplest relation between sound power level and sound pressure level is found for a free-field, non-directional
sound source, as given by the following equation:
Lp = Lw - 20 log10r - k + T
Lp = sound pressure level (dB) re 20 µPa
Lw = sound power level (dB) re 10-12 watts
r = distance from the source in meters or feet
k = 11.0 dB for metric units and 0.5 dB for English units
T = correction factor for atmospheric pressure and temperature (dB) (since most industrial noise problems are concerned with air at or near
standard conditions, T is usually negligible and , therefore, equals 0)
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Directional Sound in a Free-Field |
If the sound is directional in a free field, the relationship between sound power level and sound pressure level becomes (T=0):
Lp = Lw - 20 log10r + 10 log10Q - k
Q = directivity factor
The Directivity Factor (Q) is a dimensionless quantity that is a measure of the degree to which sound emitted by a source is concentrated in
a certain direction rather than radiated uniformly in a spherical pattern. Directivity factors for radiation patterns associated with various
surfaces surrounding a sound source are shown below. Basically, each radiation pattern is a portion of a spherical radiation pattern; that
is, a fraction of the area of a sphere (4pr2). The relationship between Lp and Lw
is also provided for each radiation pattern, as simplified from the previous
equation.
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Spherical Radiation
Q = 1
Lp = Lw - 20 log10r - 1 (r in feet)
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1/2 Spherical Radiation
Q = 2
Lp = Lw - 20 log10r + 2 (r in feet)
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1/4 Spherical Radiation
Q = 4
Lp = Lw - 20 log10r + 5 (r in feet)
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1/8 Spherical Radiation
Q = 8
Lp = Lw - 20 log10r + 8 (r in feet)
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Example: Consider a point source having a Lw of 110 dB, located outdoors on the ground. The sound
pressure level at a distance of 20 feet from the source would be calculated as follows (since the source is located on the ground, the
equation for hemispherical radiation from a point source is used):
Lp = Lw - 20 log10r + 2, therefore:
Lp = 110 dB - 20 log10(20) + 2 = 86 dB
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Unknown Sound Pressure Level |
In most situations, the sound power level is unknown. However, we can measure the sound pressure level at a point in
the far field, and relate it to another point further from the source. For sound radiation from a point source in a free field, the
following relationship applies:
Lp1 - Lp2 = 20 log10 (r2/r1)
Lp1 = sound pressure level at point 1
Lp2 = sound pressure level at point 2
r1= distance from source to point 1
r2 = distance from source to point 2.
- The distances r2 and r1 can be in any units of length (for example, feet, meters), so long as they are the same
units, since the argument of a logarithm must always be dimensionless.
- Note the particular case when r2 = 2r1:
Lp1 - Lp2 = 20 log10 (r2/r1) = 20 log (2) = 6 dB
- This is the familiar inverse-square law (decrease of 6 dB by doubling the distance).
Example:
Consider a sound pressure level of 100 dB at a distance of 10 feet. The sound pressure level at a distance of 30 feet
from the source would be calculated as follows:
Lp2 = Lp1 - 20 log10 (r2/r1) = 100 - 20 log (30/10) = 90.5 dB
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