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# Using the Decibel - Part 2: Expressing Power as an Audio Level

## 6/4/2008 2:37 PM EDT

The Decibel in Acoustics
2.7 The Decibel in Acoustics - LP, LW, and LI
In acoustics, the ratios most commonly encountered are changes in pressure levels. First, there must be a reference. The older level was 0.0002 dyn/cm2, but this has recently been changed to 0.00002 N/m2 (20 µN/m2). Note that 0.0002 dyn/cm2 is exactly the same sound pressure as 0.00002 N/m2. Even more recently the standards group has named this same pressure pascals (Pa) and arranged this new unit so that:

20 µPa = 0.0002 dyn/cm2.               (2-20)

This means that if the pressure is measured in pascals:

LP = 20log(x Pa/20 µPa)               (2-21)

If the pressure is measured in dynes per square centimeter (dyn/cm2), then:

LP = 20log(x dyn/cm2)/(0.0002 dyn/cm2).               (2-22)

The root mean square sound pressure P can be found by:

Prms = 2πfAρc               (2-23)

where,
Prms is in pascals,
f is the frequency in Hertz (Hz),
A is particle displacement in meters (rms value),
ρ is the density of air in kilograms per cubic meter (kg/m3),
c is the velocity of sound in meters per second (m/s),
ρc = 406 RAYLS and is called the characteristic acoustic resistance (this value can vary),

or

LP = 20log(Prms/20 µPa).               (2-24)

These are identical sound pressure levels bearing different labels. Sound pressure levels were identified as dB-SPL, and sound power levels were identified as dB-PWL. Currently, LP is preferred for sound pressure level and LW for sound power level. Sound intensity level is LI:

LI = 10log((x W/m2)/(10-12 W/m2)).               (2-25)

At sea level, atmospheric pressure is equal to 2116 1b/ft2. Remember the old physics lab stunt of partially filling an oil can with water, boiling the water, and then quickly sealing the can and putting it under the cold water faucet to condense the steam so that the atmospheric pressure would crush the can as the steam condensed, leaving a partial vacuum?

1 Atm = 101,300 Pa

therefore,

LP = 20log(101,300 / 0.00002)
= 194 dB.

This represents the complete modulation of atmospheric pressure and would be the largest possible sinusoid. Note that the sound pressure (SP) is analogous to voltage. An LP of 200 dB is the pressure generated by 50 lb of TNT at 10 ft. Table 2-3 shows the equivalents of sound pressure levels.

Table 2-3. Equivalents of Pressure Levels

For additional insights into these basic relationships, the Handbook of Noise Measurement by Peterson and Gross is thorough, accurate, and readable.

2.8 Acoustic Intensity Level (LI), Acoustic Power Level (LW), and Acoustic Pressure Level (LP)

Acoustic Intensity Level, LI
The acoustic intensity Ia (the acoustic power per unit of area - usually in W/m2 or W/cm2) is found by:

LI = 10log(x W/m2/10-12 W/m2               (2-26)

LI = 10log(1.0 W/m2/10-12W/m2

= 120 dB.

Acoustic Power Level, LW
The total acoustic power can also be expressed as a level (LW):

LW = 10log(Total acoustic watts/10-12 W).               (2-27)

Acoustic Pressure Level, LP
To identify each of these parameters more clearly, consider a sphere with a radius of 0.282 m. (Since the surface area of a sphere equals 4πr2, this yields a sphere with a surface area of 1 m2.) An omnidirectional point source radiating one acoustic watt is placed into the center of this sphere. Thus, we have, by definition, an acoustic intensity at the surface of the sphere of 1 W/m2. From this we can calculate the Prms:

Prms = √(10 Wa x ρc)               (2-28)

where,
Wa is the total acoustic power in watts,
ρc equals 406 RAYLS and is called the characteristic acoustic resistance.

Knowing the acoustic watts, Prms is easy to find:

Prms = √(10 Wa x 406)

= 20.15 Pa.

Thus, the LP must be:

LP = 20log(20.15 Pa/20 µPa)

= 120 dB.

And the acoustic power level in LW must be:

LW = 10log(1 W/10-12 W

= 120 dB.

Thus, the LP, LI, and LW at 0.282 m are the same numerical value if the source is omnidirectional, see Fig. 2-4.

sharps_eng

5/15/2011 11:57 AM EDT

I am finding this series a useful recap. Although I started in pro-audio I was seduced by embedded processors and digital video, but audio was definitely my first love - it's so damn analog!
The thrill was that to achieve barely acceptable SNRs of -94dB when working with 26dB of headroom took everything we had - the best brains, the best parts and then some. But we did it, and othing much has changed in terms of the net quality delivered, especially in a digital world where we can mix at 96kHz and 24 to 96bits if we want but the compression artefacts in the listeners' source material will reduce the SNR to 12 bits, or worse if we include class H amplification side-effects.
Hopefully the incease in available bandwidth will increase the amount of releases on 16bit linear PCM source, just to get rid of that quantisation weebling and psychacoustic algorithmic breathing!
Hifis still can't reproduce a piano in full voice, let alone a drum kit - maybe a good system a Marshall stack heard from way off.

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