Exhaust- Fundamentals of Acoustics

Sound vs. Noise

The phenomenon of sound in a fluid essentially involves time dependent changes of density, with which are associated time-dependent changes of pressure, temperature and positions of the fluid particles. At levels of sound experienced in everyday life, the changes of density, pressure and temperature are extremely small in relation to their mean values in the absence of sound.

 If one could observe the motion of the fluid particle in a sound wave generated by a sound source operating in a largely non-reflecting environment, such as the air above a hay field, one would see it moving to and fro along the direction of propagation. Consequently, sound waves in fluids are longitudinal waves, unlike water waves.

 Sound is both a physical phenomenon and the sensation of hearing. One hears sound but the pressure waves still exist even if there is no listener. Noise is unwanted sound. One person's sound can be another person's noise. Unlike sound, noise can only exist in the presence of a listener. The lowest sound level that an average person can hear is about 20 decibel [dB]. The human ear can tolerate sound levels to 100 dB, and in some instances even higher for short durations without lasting hearing damage. Sound levels in excess of 90 dB over an extended period can cause permanent loss of hearing.

 Sound pressure, sound power, and sound intensity are the three main characteristics used to represent sound. The sound pressure level, sound power level, and sound intensity level are the same except converted to the dB scale instead of physical units.

Sound Pressure Level

Sound pressure is the most commonly measured physical attribute of sound. Units are dB but pressure units are standard pressure units, pascals [Pa].

Pref = 0.00002 Pa. The reference pressure represents approximately the threshold of normal hearing at 1 kHz. The reference pressure represents 0 dB SPL. The sound pressure is the most representative characteristic of sound when it comes to the sound level that a human hears because the pressure in the sound wave is what causes the eardrum to vibrate causing the act of hearing. Many engine manufacturers will rate their engine exhaust sound in units of pressure.

Pressure Distance Law

When sound propogates through a fluid it scatters in many directions causing it’s energy and pressure to decrease. The sound pressure level at a new distance can be calculated from a relationship called the Pressure Distance Law (using physical units, not dB):

Where:

P1 = pressure at known distance [Pa]

P2 = pressure at distance of interest [Pa]

D1 = known distance [ft, m, in…]

D2 = distance of interest [ft, m, in…]

 

The pressure distance law applied to the dB scale for sound pressure level is:

Where:

LP1 = pressure level at known distance [dB]

LP2 = pressure level at distance of interest [dB]

D1 = known distance [ft, m, in…]

D2 = distance of interest [ft, m, in…]

Sound Power Level

Units are standard power units, watts [W].

Wref = 10-12 W. The reference power corresponds to the power passing through 1 m2 of a plane wave of intensity Iref. Sound power level is a representation of the total sound energy flow. It represents the amount of energy that an object or sound source is emitting over a given amount of time. Even though most engine manufacturers will rate their engine exhaust in units of pressure, many of them state the power as well, or sometimes they state the power only.

“The sound power level is the cause, the sound pressure level is the effect.”

Converting sound power to sound pressure

See “Divergence.”

Sound Intensity Level

Units are power over an area, W/m2.

Iref = 10-12 W/m2. Sound intensity is a measure of the power of sound projected over an area. If you can imagine a point source of sound that is emitting a given sound power, and then think of an imaginary spherical surface surrounding that point source. The intensity level of sound projected on the spherical surface would be equal to the power level of the point source divided by the surface area of the sphere. Therefore the sound intensity level decreases as the sphere’s diameter increases and vice versa. The sound intensity level would be equal to the power level of the point source when the area of the sphere is equal to one. The sound intensity level can increase above the source power when the diameter decreases enough that the surface area becomes <1. Sound intensity level is not widely used in the natural gas compression industry to rate engine exhaust sound levels.

Octave Bands

It is common for frequencies to be broken up into small ranges called bands. On the most basic scale there are 8 bands, hence “octave.” In the octave band scale the highest frequency in the range is two times the lowest frequency and the “center frequency” is the average of the highest and lowest frequencies. The center frequency is usually the frequency that the sound level is reported on. The standard octave bands are called the 1/1 octave bands and there is also another scale called the 1/3 octave bands. The 1/3 octave bands basically break each of the 1/1 octave bands into 3 separate bands. The 1/1 and 1/3 octave bands are broken down in the following tables.

Octave Band Number

Low

Center Frequency (Hz)

High

A-weight

B-weight

C-weight

 

1/3 Octave Band Number

Low

Center Frequency (Hz)

High

A-weight

B-weight

C-weight

0

22

31.5

44

-39.4

-17.1

-3

 

14

22

25

28

-44.7

-20.4

-4.4

1

44

63

88

-26.2

-9.3

-0.8

 

15

28

31.5

35

-39.4

-17.1

-3

2

88

125

176

-16.1

-4.2

-0.2

 

16

35

40

44

-34.6

-14.2

-2

3

176

250

353

-8.6

-1.3

0

 

17

44

50

57

-30.2

-11.6

-1.3

4

353

500

707

-3.2

-0.3

0

 

18

57

63

71

-26.2

-9.3

-0.8

5

707

1000

1414

0

0

0

 

19

71

80

88

-22.5

-7.4

-0.5

6

1414

2000

2825

1.2

-0.1

-0.2

 

20

88

100

113

-19.1

-5.6

-0.3

7

2825

4000

5650

1

-0.7

-0.8

 

21

113

125

141

-16.1

-4.2

-0.2

8

5650

8000

11300

-1.1

-2.9

-3

 

22

141

160

176

-13.4

-3

-0.1

9

11300

16000

22500

-6.6

-8.4

-8.5

 

23

176

200

225

-10.9

-2

0

 

 

 

 

 

 

 

 

24

225

250

283

-8.6

-1.3

0

 

 

 

 

 

 

 

 

25

283

315

353

-6.6

-0.8

0

 

 

 

 

 

 

 

 

26

353

400

440

-4.2

-0.5

0

 

 

 

 

 

 

 

 

27

440

500

565

-3.2

-0.3

0

 

 

 

 

 

 

 

 

28

565

630

707

-1.9

-0.1

0

 

 

 

 

 

 

 

 

29

707

800

880

-0.8

0

0

 

 

 

 

 

 

 

 

30

880

1000

1130

0

0

0

 

 

 

 

 

 

 

 

31

1130

1250

1414

0.6

0

0

 

 

 

 

 

 

 

 

32

1414

1600

1760

1

0

-0.1

 

 

 

 

 

 

 

 

33

1760

2000

2250

1.2

-0.1

-0.2

 

 

 

 

 

 

 

 

34

2250

2500

2825

1.3

-0.2

-0.3

 

 

 

 

 

 

 

 

35

2825

3150

3530

1.2

-0.4

-0.5

 

 

 

 

 

 

 

 

36

3530

4000

4400

1

-0.7

-0.8

 

 

 

 

 

 

 

 

37

4400

5000

5650

0.5

-1.2

-1.3

 

 

 

 

 

 

 

 

38

5650

6300

7070

-0.1

-1.9

-2

 

 

 

 

 

 

 

 

39

7070

8000

8800

-1.1

-2.9

-3

 

 

 

 

 

 

 

 

40

8800

10000

11300

-2.5

-4.3

-4.4

 

 

 

 

 

 

 

 

41

11300

12500

14140

-4.3

-6.1

-6.2

 

 

 

 

 

 

 

 

42

14140

16000

17600

-6.6

-8.4

-8.5

 

 

 

 

 

 

 

 

43

17600

20000

22500

-9.3

-11.1

-11.2

 

Notice that there are A, B, and C-weight columns in the above tables. These are derating values that can be applied to sound pressure levels at the corresponding frequencies and will be discussed in the decibel weighting section.

Decibel Weighting

The human ear is not equally sensitive to all frequencies. At 3000 Hz 90 dB sounds much louder than it does at 500 Hz. There are three basic frequency weighting scales (A, B, and C) with the “A” scale being more heavily weighted against lower frequencies. The “A” scale (dB(A)) approximates the response of the human ear and is widely used.

Adding decibels

Decibel values can be added using the following formula. This applies to sound power, pressure, and intensity levels.

Divergence

As one gets farther away from a sound source, the intensity of noise becomes less due to the spreading of sound waves. This is called divergence and it follows an inverse square relationship with distance. The pressure can be determined from the power level radiated from the source in association with a distance to the point of interest.

Where:

P = sound pressure [Pa]

D = directivity coefficient (1 for spherical divergence, 2 for hemi-spherical divergence)

ρ = density of fluid [kg/m3]

c = speed of sound [m/s]

N = sound power [W]

r = distance from power source [m]

Spherical divergence

Spherical divergence (D = 1) is used when a source is radiating into a free field with very few obstructions or reflective surfaces. Imagine a point source radiating in all possible directions creating a sphere around the source. A simplified way to calculate the sound pressure level due to a power source radiating spherically is through the following equation:

Where:

r = distance from source [ft]

Hemisperical divergence

Hemispherical divergence (D = 2) is used when a source is radiating into a free field from the surface of a plane toward only one side of the plane. This, in essence, creates a hemisphere of sound. The ground can be thought of as the “plane” that causes the sound to radiate into only half of the sphere. Therefore most industrial applications shall be considered hemispherical divergence. A simplified way to calculate the sound pressure level due to a power source radiating hemispherically is through the following equation:

Feedback and Knowledge Base