Chapter One: An Acoustics Primer
6. What is Amplitude?  page 4
Decibels: Definition, Decibels that measure power and intensity
While power is measured in watts, a widelyused acoustic measurement for relative amplitude, power, intensity, sound pressure at the listener, and comparative voltage from a microphone is the decibel (dB). Named in honor of Alexander Graham Bell, the measurement was derived from a scale originally used to determine signal loss in telegraph and telephone lines which resulted in the bel. To mitigate fractional values of the larger bel, a decibel, which is 1/10 of a bel became the standard. A decibel is a logarithmic measurement that reflects the tremendous range of sound intensity our ears can perceive and closely correlates to the physiology of our ears and our perception of loudness. There are many different forms of decibel measurement, and it is not always clear which method of computation is being used, although a few labels exist, and you may see such qualifiers that suggest the method or reference value, such as dBm, dBV, dB SPL, dB SIL, dB SWL, dB FS, dBu, dBVU, or even scale weightings such as dB(A) and dB(C) which we use on our sound level meters and will examine later. You may also see subscripts, such as L_{W} or L_{I} that hint at things such as power (in watts) and intensity.
Ambiguity abounds in dB labeling, so take care you know what is being measured and the reference benchmarks being used. The SI (International System of Units), ISO and others do not recognize any of the suffixes above for decibels, and in literature, the letter 'p' is often used in dB equations to represent power, pascals or pressure, which are NOT interchangeable, so mind your 'p's' if not 'q's'.
In order to get started, a quick look at logarithms (often misspelled by us musicians as logarhythms) is in order.
A logarithm primer 
Be not intimidated by calculating logarithmswith cheap calculators to do the math (one previously used log tables), just a simple understanding of how they work is all that is necessary for decibel calculations (if you turn your iPhone or Android phone horizontally while using the onboard calculator, it will be happy to compute complex logs for you, but just be sure to use the log_{10} function, not natural log (ln)). Log_{10 }x can be thought of as "what power of 10 will result in x." For example, Log_{10} 100 = 2 because 10^{2 }= 100. Decibels are often used to measure very minute values, which can also be expressed by logs of decimals numbers or their negative power equivalents e.g. x^{y}. For example, log_{10 } 0.1 = 1 and log_{10} 0.000000000001 = 12 because 10^{12} = 0.000000000001, a value we will use for our threshold of hearing measurement below (a more compact way of expressing 0.000000000001 is 10^{12}^{}, so log_{10} 10^{12} = 12. If a log is expressed y = log_{10} x, then 10^{y } = x. 
A decibel is a measurement used to compare the ratio of power, intensity or amplitude between two acoustic sounds or electronic signals. The ratio (R) of two signals expressed by their power in watts (W1 and W2) is:
A doubling of power equals an increase of +3dB. When we study filters later on, you will notice that a filter cutoff frequency is defined as the halfpower point, which is calculated as –3dB.
Decibels that measure power and intensity
There are many different types of decibel measurements, so for the purpose of clarity, the above form, which measures power or intensity, is called dBm when a fixed reference value is used for the denominator. For the purpose of having a standardized absolute measurement of power in an electronic circuit (i.e., a comparison not to another signal, but to an industryfixed value), the nominal reference wattage (W2) has been defined as 1 milliwatt (0.001 watt). In absolute terms, a 1watt signal, which has 1,000 times the power of the reference wattage, will be 30 dB, computed below:
dBm=10 log_{10} (1 watt/.001 watt) dBm =10 log_{10} (1000) dBm=10 x 3 [because log_{10} 1000 = 3] dBm=30 
dBm is the form most commonly used to evaluate power in audio circuits.
For acoustic measurement of the total power radiated in all directions (referred to as Sound Power Level, or SWL), the same formula is used with a reference level correlating to threshold of audibility that we mentioned earlier, 10^{12} watts (also called a picowatt). Note how loud a sound with 1 watt of total acoustic energy might be at 120 dB SWLit is the reported sound of a typical jet plane at 500 feet (whaaat, those four engines produce only 1 acoustic watt of sound? Yeah.).
dB SWL=10 log_{10} (1 watt/10^{12} watts) dB SWL =10 log_{10} (10^{12}) [that's some nifty math there] dB SWL=10 x 12 [because log_{10} 10^{12} = 12] dB SWL=120 
Since intensity (I) at a fixed distance of measurement is directly proportional to power, a similar measurement can be made for intensity using a reference value of a picowatt per meter squared or I=10^{12} W/m^{2} in place of I_{ref} below to make it a Sound Intensity Level (SIL). As a reminder, acoustic intensity is the measurement of power (or the rate at which energy passes) through a defined area perpendicular to the direction of wave propagation. So we need both a power measure (watts) over an area measurement (meters^{2})
A realworld example of intensity measured in decibels 
As you recall from the chart on the previous page, intensity is a measure of power present over or through an area, so let's figure out the acoustic intensity in dB SIL present at the surface of a loudspeaker with a cone radius of 0.5 meters receiving 1 watt of power. As stated, the common reference level for I_{ref}_{} is 10^{12} W/m^{2}.* Note the example below contains some rounding for clarity. STEP 1: Find the area (A) of the speaker cone. The area of a circle is πr^{2}, so the cone area = π(0.5^{2) }= π0.25 ≈ 0.78 square meters. STEP 2: Compute the sound intensity in watts per meter squared by dividing the applied power of 1 watt by the cone area of 0.78 m^{2}. So 1/0.78 W/m^{2} ≈ 1.28 W/m^{2} gives us the sound intensity at the surface of the speaker. STEP 3: Compute the dB SIL using the STEP 2 sound intensity of 1.28 W/m^{2} and the SIL reference value of 10^{12} W/m^{2} plugged into the dB SIL formula above: So, 1 watt of power in terms of intensity I at the loudspeaker cone gives us an approximately earsplitting 121 dB SIL, very close to the 120 dB SWL above using 1 watt signal and the picowatt referenceit would be even closer without my rounding errors. As we moved farther away from the speaker and the sound field spreads out, this measurement would change dramatically at the surface of the sound sphere as our STEP 1 area increased geometrically. BTW, you might be wondering why you'd buy a 500 watt amplifier if you can get 121 dB intensity out of a honking large speaker with only 1 watt of power. The sad fact is speakers are extremely inefficient at converting electrical power into acoustic power, so alas, only a small fraction of this energy is transferred. *Thanks to David Howard/Jamie Angus Acoustics and Psychoacoustics, 4th ed. 2009 for this example idea. 
Decibels that measure amplitude, microphone voltage, sound pressure, and digital systems
Amplitude can also be measured in decibels with a slight modification that keeps it proportional to the measure for power above using the standard reference values for watts and pressure. With power being proportional to amplitude squared, we can create the formula (W=watts, p=pressure): 10 log_{10} (W/W_{ref} ) = 10 log_{10} (p/p_{ref} )^{2 }= 20 log_{10} (p/p_{ref} ), so the big difference is the multiplier of 20 and the units of measurement. For sound pressure, the standard unit is the pascal, which we discussed earlier is equivalent to a N/m^{2}. A micropascal = 0.000001 of a pascal.
Note the only difference is a multiplication factor of '20' vs. '10' for power and intensity measurements. By comparing this formula to the one for power ratios above, the relationship between amplitude, power and intensity becomes clear. Power and intensity are usually proportional to the square of amplitude, and the formula above will give identical results for the same amplitude ratios. Using the formula above, we see a doubling of amplitude from one source to another equals an increase of +6 dB as shown below:
Sound Pressure Level or SPL
As mentioned on the page before, this is the one of the most common acoustic field measurements. SPL measures a current sound against a predetermined value of the threshold of audibility, mentioned above for power, but now expressed as a sound field value of
20 μPa (20 micropascals (μPa),
1 micropascal = 10^{6} pascals). This absolute measurement is referred to as the soundpressure level (SPL). It
gives us a means of generalizing relative loudness of common acoustic sources.
(Note that the "dB" is followed by "SPL" to indicate this mode of measurement.)
The logarithmic scale from the threshold of hearing to the threshold of pain, expressed as intensity,
ranges from 0.00002 N/m^{2} (equivalent of 20 μPa) to 200 N/m^{2}, or about 120130 dB
SPL, at which point the entire body, not just the ears, sense the vibrations. While every 6 dB SPL represents a doubling of amplitude, a nonexact ruleofthumb is that every 10 dB increase is a doubling of perceived loudness (although this is mitigated by other factors such as frequency content, to be covered later).
Threshold of Pain: In preparing this article, it quickly became apparent that no standard for the threshold of feeling or the threshold of pain has been established. In the references consulted, the threshold of pain ranges from 120 dB SPL to 140 dB SPL, which is a huge variation of opinions and points out the differences between acoustic and psychoacoustic measurement. Younger people also have more effective protection mechanisms and so can better tolerate louder sounds — surprise!. 
If we accept 130 dB as the threshold of pain, then humans hear sounds that range from the smallest perceptible intensity to those that are 10,000,000,000,000 as loud or 10 watts/m^{2}. Both the dB and dB SPL scales reflect the incredible discrimination of human hearing, our most sensitive sense by far.
FACTOID: The SPL pressure reference value of 20 μPa and the SIL picowatt/m^{2} reference value were chosen to keep the proportionality of SIL and SPL as such: Intensity in watts/m^{2} = Pressure in pascals^{2} when both these reference values are used. And this makes the two correlated columns of the benchmark chart below possible. 
Here are some vague benchmarks (which of course depend on many factors, including the listener’s distance from the sound or your particular model of vacuum cleanerif you even have a vacuum cleaner).
Source  Intensity (watts/m^{2})  dB SPL 
Threshold of pain  10  130 
Jet takeoff from 500 ft.  1  120 
Mediumloud rock concert 50 ft.  .1  110 
Circular saw 3 ft.  .01  100 
New York subway from platform  .001  90 
Jackhammer from 50 ft.  .0001  80 
Vacuum cleaner from 10 ft.  .00001  70 
Normal conversation 5 ft.  .000001  60 
Light traffic from 100 ft.  .0000001  50 
Soft conversation 5 ft.  .00000001  40 
Whisper from 5 ft.  .000000001  30 
Average household silence  .0000000001  20 
Breathing .5 ft.  .00000000001  10 
Threshold of hearing in young  .000000000001  0 
Amazing Factoid: The Ben and Jerry's Ice Cream Company recently funded research on a freonfree freezer that uses sound waves pumped in at an astonishing 190 decibels to compress the air enough to bring the temperature down to 0 degrees. MORE AMAZING FACTOID!: dB Drag Racing is a growing sport in which competitors attempt to produce the loudest sound possible inside a vehicle, which must also be able to run. Wikipedia reports the highestamplitude incar sound produced to date is 180 dB SPL6 times the perceived loudness of a jet plane takeoff at 500 feet, and about the same SPL as a police flashbang grenade. TOTALLY, TOTALLY AMAZING FACTOID!!: The pistol shrimp creates a collapsing cavitation bubble by quickly snapping its claw that produces an acoustic pressure of 80 kPa (80,000 pascals) and stuns its prey with a whopping 218 decibels of sound pressure (it may also cause sonoluminescence)! Plug that into our above SPL formula using the 20 μPa reference! 
Signals from microphones, most of which seek to accurately transform changes in SPL to proportional changes in voltage (V), can also be measured by the same method. If one were to change the miking distance to the sound source, the voltage differences could be measured as follows:
If measured properly, halving the distance of the mic to the source, thanks to the inverse square law, should double the voltage produced by the microphone, giving a +6 dB increase in amplitude (which, if you’ve been reading closely, also produces four times the intensity). For a standardized comparison of voltages, 0.775 volts is used as the reference level for 0 dB.
Decibels in the digital age: dBFS (dB Full Scale Digital)
As digital circuitry evolved, a new form of dB evolved in the midseventies designated dBFS for fullscale digital. Unlike analog systems, which have a certain degree of "headroom" above their ideal peak level, digital systems do not. There is an absolute maximum peak amplitude quantity that is determined by the number of bits per digital sample. The maximum digital amplitude would be designated as 0 dBFS. The rest of the dBFS measurement is scaled to the dynamic range of the sample size, so that a signal averaging say 50% of the maximum value would have a level of 6 dBFS, very much as 6 dB SPL would be a halving of the averaged sound pressure level or voltage differential.
Though the logic of digital audio reproduction is absolute, most manufacturers of digital audio equipment and software such as digital audio workstations (Pro Tools, Logic, Digital Performer) realize that their users either long for the old days of analog VU meters and headroom, or the distorted sound coloring that ensued by overloading tape heads and analog circuits beyond 0 dBVU (or just want way more sound beyond the legal limit because they feel entitled to it). They therefore calibrate their software so that the '0' on the fader or meter means approximately 12 to 18 dBFS, and allow users to crank it up another +6 to +18 dB to reach true 0 dBFS as demonstrated below. In fact, many audio recording gurus, such as the legendary engineer Bob Katz in Mastering Audio, tout the ideal music mixing sweetspot at 12 to 18 dBFS average (so RMS, not peak), which is equivalent to 0 dBVU of the past. dBFS meters will often have a ∞ (negative infinity) sign at absence of signal or silence (see right meter below).
MOTU Digital Performer Example: Note the difference between the DP mixing board fader indications on the left (old skool analog SPL indications, actually pertaining to the fader's level, not the signal meter next to it) and the METER BRIDGE meter, which does give the accurate dBFS level of the track's signal. At a level of close to '0' on the mixing board's meter, the true dBFS is more like 6 or 7 dBFS on the METER BRIDGE.

Summary
We have looked at two basic types of dB measurement, one for power and intensity, and the other for amplitude, SPL and voltage. Several other weighted dB scales, such as dBA are used for specific purposes, such as more closely mirroring the way we hear, but this will be discussed in further detail in the psychoacoustics sections. You may also run into the terms sones and phons at some point in your studiesthese are also psychoacoustic measurements i.e. designed to factor in the way we hear.