## The Decibel Scale for Sound Intensity and PressureRobert F. Port

Intensity
Power is a measure of work per unit of time, and is measured in watts. For example, audio amplifiers are described by their maximum power output (e.g., ``30 watts peak power'').

In studying hearing or speech our interest is usually not in the total power of a sound source, but in the power per unit of area, the Intensity, as in  watt/m2. This is because the human ear offers a fixed area (the ear drum) for detection of sounds in the environment. Intensity is the amount of energy transmitted per second over an area of a square meter.

The sound intensities that human ears are sensitive to are very small compared to a watt, but still very wide between the weakest sound detectible and the strongest that does not cause pain (or damage the ear). Since the watt itself is far too large, the reference level of intensity that all others are compared to is, by convention, 10 -12 watts/m 2 . This intensity was chosen as the standard reference level because it is approximately the weakest intensity of a pure tone at 1000 Hz that can be detected by human listeners, that is, an auditory threshold.  The most intense audio signal that is not painful is roughly  10 watts/m2. Since the most intense is 10 12 or 1,000,000,000,000  (a trillion times) larger than the threshold level sound, it clearly makes sense to use a logarithmic scale to discuss sound intensity.

The Bel and Decibel
For this reason, the Bel (named after A. G. Bell) was defined this way,

number of  Bels =  log10  Ix / Ir
where is Ix the signal being described and  Ir is a signal used as a reference level. The Bel can be used to compare the relative intensities of any two signals, e.g., the input and output of an acoustic filter. When a relatively `absolute' description of the intensity of a signal is needed (like if you want to say `how loud' the background noise in a room is),  the reference level, Ir , is  taken to be 10 -12 watts/m 2 . This measure is said to be an  Intensity Level or IL.

But each incremental unit in the Bel scale, from 1 to 2 or from 5 to 6, corresponds to a factor of 10 in intensity. As we said, the total range that is relevant for human hearing is thus 12 Bels. But a smaller unit would be more manageable when most practical comparisons are made. Thus the decibel, or dB was defined as one tenth of a Bel:

Intensity Level in dB  = 10 log 10  Ix / Ir

So the weakest audible sound is roughly 0 dB IL and a whispered voice is 1000 times more intense or 30 dB higher. A loud rock concert, close to the threshold of pain might have an intensity ratio of  1012 , or 120 dB IL (relative to the standard reference level).

Pressure

It turns out that a measure of sound pressure is often more useful than intensity. (For one thing, microphones basically `measure' sound pressure, and instantaneous pressure is the `amplitude' axis in a typical waveform plot.) But sound pressure is proportional to the square root of the intensity.  And conversely, intensity is proportional to the pressure squared. That is,  I = P2 .   Thus,

n dB = 10 (Px / Pr )2

Now since log ab = b log a , it follows that

n dB = 20 (Px / Pr)

When pressure is measured in dB, with respect to the standard reference level mentioned above, it is called `dB sound pressure level' or dB SPL. This is probably the most common `absolute' measure of sound level. And is closest to what most lay people think the term ``dB'' means.

The following table (based on Moore, 1989) shows some relationships between intensity ratios, pressure ratios and the standard reference level. Sound levels in dB SPL are expressed relative to a reference intensity level of   10 -12 W/m 2.

 Sound level Intensity ratio Pressure ratio Example dB SPL I / Ir P / Pr 140 1014 10 7 Gunshot at close range 120 10 12 10 6 Loud rock show 100 10 10 10 5 Shouting at close range 80 10 8 10 3 Busy city street 70 10 7 3.16 x 10 3 Normal conversation 50 10 5 316 Quiet conversation 30 10 3 31.6 Soft whisper 20 10 2 10 In the woods at night 6.5 4.5 2.1 Mean absolute threshold at 1 kHz 0 1 1 Reference level