The Decibel

As stated in the preceding paragraphs, perceived levels of sound change according to the logarithm of the actual sound power level. The same is true for various other phenomena, too, such as visiblelight intensity and radio-frequency signal strength. Specialized units have been defined to take this into account.

The fundamental unit of sound-level change is called the decibel, symbolized as dB. A change of +1 dB is the minimum increase in sound level that you can detect if you are expecting it. A change of -1 dB is the minimum detectable decrease in sound volume, when you are anticipating the change. Increases in volume are given positive decibel values, and decreases in volume are given negative decibel values.

If you aren’t expecting the level of sound to change, then it takes about +3 dB or -3 dB to make a noticeable difference.Changes in intensity, when expressed in decibels, are sometimes called gain and loss. Positive decibel changes represent gain, and negative decibel changes represent loss. The sign (plus or minus) is usually absent when speaking of changes in terms of decibel gain or decibel loss. If you say that a certain system causes 5 dB of loss, you are saying that the gain of that circuit is -5 dB.

Calculating Decibel Values

Decibel values are calculated according to the logarithm of the ratio of change. Suppose a sound produces a power of P watts on your eardrums, and then it changes (either getting louder or softer) to a level of Q watts. The change in decibels is obtained by dividing out the ratio Q/P, taking its base- 10 logarithm (symbolized as log10 or simply as log), and then multiplying the result by 10. Mathematically:

dB = 10 log (Q/P)

As an example, suppose a speaker emits 1 W of sound, and then you turn up the volume so that it emits 2 W of sound power. Then P = 1 and Q = 2, and dB = 10 log (2/1) = 10 log 2 = 10 × 0.3 = 3 dB. This is the minimum detectable level of volume change if you aren’t expecting it: doubling of the actual sound power!

If you turn the volume level back down again, then P/Q = 1/2 = 0.5, and you can calculate dB = 10 log 0.5 = 10 × −0.3 = −3 dB.

A gain or loss of 10 dB (that is, a change of +10 dB or −10 dB, often shortened to 10 dB) represents a 10-fold increase or decrease in sound power. A change of 20 dB represents a 100-fold increase or decrease in sound power. It is not unusual to encounter sounds that vary in intensity over ranges of 60 dB, which represents a 1,000,000-fold increase or decrease in sound power!

Sound Power in Terms of Decibels

The preceding formula can be worked inside out, so that you can determine the final sound power, given the initial sound power and the decibel change. To do this, you use the inverse of the logarithmic function, symbolized as log−1 or antilog. This function, like the logarithmic function, can be performed by any good scientific calculator, or by the calculator program in a personal computer when set to scientific mode.

Suppose the initial sound power is P, and the change in decibels is dB. Let Q be the final sound power. Then:
Q = P antilog (dB/10)
As an example, suppose the initial power, P, is 10 W, and the perceived volume change is −3 dB. Then the final power, Q, is equal to 10 antilog (−3/10) = 10 × 0.5 = 5 W.

Decibels in the Real World

Sound levels are sometimes specified in decibels relative to the threshold of hearing, defined as the faintest possible sound that a person can detect in a quiet room, assuming his or her hearing is normal. This threshold is assigned the value 0 dB. Other sound levels can then be quantified as figures such as 30 dB or 75 dB.

If a certain noise has a loudness of 30 dB, that means it’s 30 dB above the threshold of hearing, or 1000 times as loud as the quietest detectable noise. A noise at 60 dB is 1,000,000 (or 106) times as powerful as a sound at the threshold of hearing. Sound-level meters are used to determine the decibel levels of various noises and acoustic environments.

A typical conversation occurs at a level of about 70 dB. This is 10,000,000 (or 107) times the threshold of hearing, in terms of actual sound power. The roar of the crowd at a rock concert might be 90 dB, or 1,000,000,000 (109) times the threshold of hearing. A sound at 100 dB, typical of the music at a large rock concert if you are sitting in the front row, is 10,000,000,000 (1010) times as loud, in terms of power, as a sound at the threshold of hearing.