Page 50 - The Master Handbook Of Acoustics

25
SOUND LEVELS AND THE DECIBEL

Ratios of powers or ratios of intensities, or ratios of sound pressure,
voltage, current, or anything else are dimensionless. For instance, the
ratio of 1 watt to 100 watts is 1 watt/100 watts, and the watt unit
upstairs and the watt unit downstairs cancel, leaving ⁄100 = 0.01, a pure
1
number without dimension. This is important because logarithms can
be taken only of nondimensional numbers.

Handling Numbers
Table 2-1 illustrates three different ways numbers can be expressed.
The decimal and arithmetic forms are familiar in everyday activity.
The exponential form, while not as commonly used, has the charm of
simplifying things once the fear of the unknown or little understood is
conquered. In writing one hundred thousand, there is a choice
5
between 100,000 watts and 10 watts, but how about a millionth of a
millionth of a watt? All those zeros behind the decimal point make it
impractical even to reproduce here, but 10 –12 is easy. And the prefix
that means 10 –12 is pico; so the power is 1 picowatt (shown later in
Table 2-4). Engineering-type calculators take care of the exponential
form in what is called scientific notation, by which very large or very
small numbers can be entered.

Table 2-1. Ways of expressing numbers.
Decimal Arithmetic Exponential
form form form
100,000 10 × 10 × 10 × 10 × 10 10 5
10,000 10 × 10 × 10 × 10 10 4
1,000 10 × 10 × 10 10 3
100 10 × 10 10 2
10 10 × 1 10 1
1 10/10 10 0
0.1 1/10 10 –1
0.01 1/(10 × 10) 10 –2
0.001 1/(10 × 10 × 10) 10 –3
0.0001 1/(10 × 10 × 10 × 10) 10 –4