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

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

This means that if the pressure is measured in pascals:

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

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

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

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

P_rms = 2πfAρc               (2-23)

where,
P_rms 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

L_P = 20log(P_rms/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, L_P is preferred for sound pressure level and L_W for sound power level. Sound intensity level is L_I:

L_I = 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,

L_P = 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 L_P 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.

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 (L_I), Acoustic Power Level (L_W), and Acoustic Pressure Level (L_P)

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

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

L_I = 10log(1.0 W/m²/10^-12W/m²

      = 120 dB.

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

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

Acoustic Pressure Level, L_P
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:

P_rms = √(10 W_a x ρc)               (2-28)

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

Knowing the acoustic watts, P_rms is easy to find:

P_rms = √(10 W_a x 406)

        = 20.15 Pa.

Thus, the L_P must be:

L_P = 20log(20.15 Pa/20 µPa)

      = 120 dB.

And the acoustic power level in L_W must be:

L_W = 10log(1 W/10^-12 W

      = 120 dB.

Thus, the L_P, L_I, and L_W at 0.282 m are the same numerical value if the source is omnidirectional, see Fig. 2-4.