The Decibel

Now that you're completely comfortable with logarithms you should have no problem with the decibel. You've probably heard the term used before to describe the intensity of sound, but never really understood what it meant. The decibel is defined as 10 times the log of sound power relative to a reference. Specifically,

dB = 10log(P/Preference),

where P is the total power of the sound and Preference is the power of the reference. It is important to note from this definition that the dB level of the sound depends on the reference value chosen. In other words, the dB has no meaning unless the reference value is known. The reference is usually indicated after the dB value. For example, dB re: 1 watt means that the reference is 1 watt (re: is to be read 'in reference to'). Actually the reference need not always be expressed as a power. A very popular reference is sound pressure level (SPL) corresponding to a pressure value of 0.0002 dynes/cm². If we are to use a pressure value as a reference we must be careful to convert pressure to power to be consistent with the definition of dB. Since power is proportional to pressure squared, this amounts to replacing the power ratio in Eq. (1) with a pressure ratio squared. And, because the log of any value squared is just 2 times the log of that value (see rule iii), we have,

dB SPL = 20log(p/0.0002 dynes/cm²).

We will discuss the reasons for the popularity the SPL reference in class.