Page 46 - Master Handbook of Acoustics

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loud sound (causing a sensation of pain) might be 10 W/m . (Acoustic intensity is acoustic power per
unit area in a specified direction.) This range of intensities from the softest sound to a painfully loud
sound is 10,000,000,000,000. Clearly, it is more convenient to express this range as an exponent,

13
−12
2
10 . Furthermore, it is useful to establish the intensity of 10 W/m as reference intensity I and
ref
express other sound intensities I as a ratio I/I to this reference. For example, the sound intensity of
ref
2
−9
3
2
−9
10 W/m would be written as 10 or 1,000 (the ratio is dimensionless). We see that 10 W/m is
1,000 times the reference intensity.
Logarithms

2
3
Representing 100 as 10 simply means that 10 × 10 = 100. Similarly, 10 means 10 × 10 × 10 =
1,000. But what about 267? This is where logarithms are useful. Logarithms are proportional
numbers, and a logarithmic scale is one that is calibrated proportionally. It is agreed that 100 equals

2
10 . By definition we can say that the logarithm of 100 to the base 10 equals 2, commonly written
log 100 = 2, or simply log 100 = 2, because common logarithms are to the base 10. The number 267
10
can be expressed as 10 to some power between 2 and 3. Avoiding the mathematics, we can use a
calculator to enter 267, push the “log” button, and 2.4265 appears. Thus, 267 = 10 2.4265 and log 267 =
2.4265. Logarithms are handy because, as Table 2-1 demonstrates, they reduce multiplication to
addition and division to subtraction.
Logarithms are particularly useful to audio engineers because they can correlate measurements to

human hearing, and they also allow large ranges of numbers to be expressed efficiently. Logarithms
are the foundation for expressing sound levels in decibels where the sound level is a logarithm of a
ratio. In particular, a sound level in decibels is 10 times the logarithm to the base 10 of the ratio of
two powerlike quantities.

Decibels

We observed that it is useful to express sound intensity in ratios. Furthermore, we can express the
intensities as logarithms of the ratios. An intensity I can be expressed in terms of a reference I as
ref
follows:

The intensity measure is dimensionless, but to clarify the value, we assign it the unit of a bel (from

Alexander Graham Bell). However, when expressed in bels, the range of values is somewhat small.
To make the range easier to use, we usually express values in decibels. The decibel is 1/10 bel. A
decibel (dB) is 10 times the logarithm to base 10 of the ratio of two quantities of intensity (or power).
Thus, the intensity ratio in decibels becomes: