DECIBELS

Decibels scales are widely used to compare two quantities. We may express the power gain of an amplifier in decibels (abbreviated dB), or we may expresss the relative power of two sound sources.  We could even compare our bank balance at the beginning with the balance at the end of the month. ("My bank account decreased 27 decibels last month.") The decibel difference between two power levels, ΔL, is  defined in terms of their power ratio W₂/W₁:

ΔL = L₂ - L₁ = 10log W₂/W₁

Although decibel scales always compare two quantities, one of these can be a fixed reference, in which case we can express another quantity in terms of this reference.  For example, we often express the sound power level of a source by using w₀=10^-12W as a reference.  Then the sound power level (in decibels) will be

L_w = 10log W/W₀

One number fact worth remembering is that the logarithm of 2 is 0.3 (actually 0.3010. but 0.3 will do).  Why is this worthwhile? Because 10log 2 - 3, doubling the power results in an increase of 3dB in the power level.

Power ratio: 2   3   4   5   10   100

Decibel Gain: 3   5   6   7   10   20

SOUND INTENSITY LEVEL

We have just seen how the strength of a sound source can be expressed in decibels comparing its power to a reference power (nearly always W₀=10¹²W).  Similarly the sound intensity level at a point some distance from the source can be expressed in decibels by comparing it to a reference intensity, for which we generally use I₀=10^-12W/m^2.  Thus the sound intensity level at some location is defined as

L₁ = 10log I/I₀

Even though they are both expressed in decibels, do not confuse sound power level, which describes the sound source, with sound intensity level, which describes the sound at some point.  The relationship between the sound intensity level at a given distance from a sound source and the sound power level of the source depends upon the nature of the sound field.

SOUND PRESSURE LEVEL

In a sound wave there are extremely small periodic variations in air pressure to which our ears respond.  The minimum pressure fluctuation to which the earcan respond is less than 1 billionth (10^-9) of atmospheric pressure.  This threshold of audibility, which varies from person to person, corresponds to a sound pressure amplitude of about 2 X 10^-5 N/m² at a frecuency of 1000 Hz.  The threshold of pain corresponds to a pressure amplitude approximately 1 million (10⁶) times greater, but still less than 1/1000 of atmospheric pressure.

The intensity of a sound wave is proportional to the pressure squared.  In other words, doubling the sound pressure quadruples the intensity.  The actual formula relating sound intensity I and sound pressure p is

I = p²/pc

where p is the density of air and c is the speed of sound.  The density p and the speed of sound c both depend on the temperature.  At normal temperatures the pc is around 410 to 420, but for ease of caculation, we often set it equal to 400.

It is useful to substitute for I from the above equation pc = 400 in L_I = 10log p²/400I_o = 10log p²/4 x 10^-10 = 20log p/2 x 10^-5. The latter expression is  defined as the sound pressure level L_p (sometimes abbreviated SPL, although L_p is prefered). 

L_p = 20log p/p₀

LOUDNESS LEVEL

Although sounds with a greater L₁ or L_p usually sound louder, this is not always the case.  The sensivity of the ear vavries with the frecuency and the equality of the sound.  Many years ago Fletcher and Munson (1933) determined curves of equal loudness level (L_L) for pure tones ( that is, tones of a single frecuency). 

These curves demostrate the relative insensitivity of ear to sound of low frecuencies at moderate to low intensity levels.  Hearing sensitivity reaches a maximum between 3500 and 4000 Hz, which is near the first resonance frecuency of the outer cana, and again peaks around 13Hz, the frecuency of the second resonance.

The contours of equal loudness level are labeled in units called phons, the level in phons being numerically equal to the sound pressure level in decibels at f = 1000Hz.  The phon is a rather arbitrary unit, however, and is not widely used in measuring sound.  Its important. however. to note the relative insensitivity of the ear to sound of low frecuency, which is one reason why weighting networks are used in sound-measuring equipment.

Sound level meters have one or more weighting networks, which provide the desired frecuency responses.  Generally three weighting networks are used; they are designated A,B, and C. The C-weighting network has an almost flat frecuency response, whereas the A-weighting network introduces a low-frecuency rolloff in gain that bears rather close resemblance to the frecuency response of the ear at low sound pressure level.  

Sources:

Rossing, Thomas D., F. Richard Moore, and Paul A. Weeler, The Science of Sound, Third Edition. Addison Wesley, 2002.