# What is the unit for sound intensity?

I hear people say "decibel" for sound intensity, but a dB is just the ratio between two sound intensity levels, so it's really not a unit.

So, if r and p represent the reference power level and measured power level, respectively (where r is a measurement of power also), then:

dB = 10log(p/r).

What are the two sounds measured in however, to achieve this ratio? In other words, what reference power level is used?

Also, since I expressed the two measurements, r and p, in power, you could say that the Watt the unit of measurement. Though one could also use voltage, amperage, or perhaps other such units.

All those units are actually related by the way, one affects the other, so it can't just be an arbitrary measurement. If you know this already then I don't want to sound patronizing, just being safe so as not to cause confusion :)

• I'm guessing you're looking to measure sound intensity. Since there are multiple aspects of sound that can be measured, it doesn't make sense to ask for a "unit for sound". Other components that are commonly measured are sound frequency (measured in Hertz, or Hz) and sound duration (measured in time units such as seconds). There are also psychoacoustic properties that are often measured, such as loudness (measured in phons) and pitch (represented most often by notes and cents relative to notes). – Todd Wilcox Mar 28 '16 at 2:28

First of all, when using or abbreviating a unit named after a person (in this case, Alexander Graham Bell), it is customary to capitalize the abbreviation, so the most respectful way to write the common measurement of the ratio between to values is dB. Capitalization is not used when the unit is spelled out with a scale prefix ("deci-", in this case), but the unit written out without a scale prefix is capitalized: "Bel".

Almost always, decibels are measured and reported based on a reference value, and the reference value is indicated in the unit abbreviation by extra characters added on after "dB".

Examples (the first one being the most common answer to your question):

• dB SPL (sound pressure level) – for sound in air and other gases, relative to 20 micropascals (μPa) = 2×10^(−5) Pa. Very frequently, dB SPL measurements of wideband sounds (like music, as opposed to a single frequency) are measured as an average across frequencies and then weighted to approximate the response of human hearing. The most common weighting is called "A-weighted". Sometimes this will be abbreviated dB(A). Note that SPL measurements are pressure and not power levels.
• dBm – electrical power relative to 1 milliwatt.
• dBu – RMS voltage relative to approximately 0.7746 V.
• dBV – voltage relative to 1 volt, regardless of impedance.
• dBFS (Full Scale) – the amplitude of a signal compared with the maximum which a device can handle before clipping occurs. This is used for levels in digital audio when samples are composed of signed integers of fixed bit depth. If the level were referenced to integer 0, different bit depths would have different level measurements. By using dBFS, levels can be directly compared between different digital formats. An absolute value of 32,767 in a 16-bit file and an absolute value of 8,388,607 in a 24-bit file would both be full scale and therefore be represented by 0 dBFS. Changing the MSB from a 1 to a 0 in each of these values would halve the total integer value and represent a drop of 6 dB, which would be a level of -6 dBFS in both cases.

Important Note: All of the above values are actually "20-log" dB measurements except the dBm and the dB SPL, which means the formula for calculating the dB based on the reference value is actually 20log(m/r) (m being measured value and r being the reference value). Power levels are scaled by 10 times the logarithm of the ratio, while intensity or voltage levels are scaled by 20 times the logarithm of the ratio. I'm not exactly sure why this is done, but I do know that it helps when calculating gain and output, etc. That's because, for example, in an electrical system power and voltage are not linearly related. If you halve the voltage into a given load, you reduce the power output to a quarter of the previous value. By using the 10-log and 20-log systems, half the voltage is a 6 dB drop and one quarter the power is also a 6 dB drop. The mixed scales make it so following gain changes through a system is much easier than if a consistent scale were used for both power and intensity.

## Edit:

I just noticed (while editing your question to make it more clear) that you specifically asked about acoustic power levels. Acoustic power is measured and reported in straight-up Watts, and is also measured in dB referenced to 10^(-12) Watts. Acoustic power is not often used in musical contexts because it doesn't relate as well to how we actually hear. One way to understand this is to note that the reference value for dB SPL is the average/typical measured threshold of hearing for humans. That means a sound pressure of -3 dB SPL should be inaudible to almost all humans and an pressure of 3 dB SPL should be barely audible to almost all humans. Sound power levels are not as immediately helpful in estimating the audibility and loudness of a sound.

The other difference between sound power and sound pressure is that sound power is meant to be the total output of a sound producing object, while sound pressure is the actual acoustic pressure measured at a particular point in space. As our ears exist at a particular point in space (or two points) and our ears respond to pressure and not power, sound pressure makes more sense for understanding how a sound will be heard. Sound power is more useful in engineering situations where acoustic power can effect physical objects other than human eardrums.

• Comprehensive answer. +1. It reminded me of the days when venues had cut offs that worked to stop bands playing too loudly. I always thought that they actually didn't work purely on sound pressure level, but certain frequencies as well, which seemed unfair! Could some of the facts in your answer contribute to this? – Tim Mar 28 '16 at 7:52
• @Tim As far as I know, all laws in the USA that regulate sound intensity are based on A weighted SPL measurements, despite the fact that audiologists and engineers typically view A weighting as a poor representation of loudness. – Todd Wilcox Mar 28 '16 at 11:18
• Here's a discussion of acoustics in terms of power levels: study.com/academy/lesson/… – Dave Mar 28 '16 at 14:01
• @Dave That's actually a discussion of intensity, not power. Notice that they use the word "intensity" and the formula that includes Watts in the unit has the Watts divided by meters squared. Watts divided by meters squared is pressure in Pascals and therefore it is intensity. The link did school me on the fact that intensity uses a 10 log scale. – Todd Wilcox Mar 28 '16 at 14:10
• @ToddWilcox The power required to shake something (e.g. some air) is force x velocity, or (pressure x area) x velocity. To eliminate the velocity from that equation, you have to use the physics of how a gas behaves, and the result is power = area x (pressure squared) / (density of air x speed of sound). The "pressure squared" term gives a factor of 20 not 10 when you convert to decibels. Humans judge one sound to be "twice as loud" as another on a log scale of pressure, not of power, which is why "sound pressure level" is more useful in acoustics than sound power level". – user19146 Mar 29 '16 at 7:31

Excellent answer already but here are some details in the physics for the mega nerds

1. Sound is basically air molecules wiggling back and forth (transversal mechanical wave in a fluid medium)
2. When neighboring air molecules bunch up they create a local pressure increase, if the pull away from each other they create a local pressure decrease
3. The air molecules have kinetic energy (want to keep moving) and potential energy (want to equalize the pressure), the two forms of energy are constantly converting into each other. So any one molecule constantly changes between having kinetic energy and potential energy as it wiggles.
4. The measure for kinetic energy is the velocity of the air molecules (which is NOT the speed of sound) and the measure for potential energy is the pressure
5. The sound intensity is simply the product of velocity and pressure. Velocity is measure in "meters per second" and pressure in Pascal or Newton per square meter. Newton is the official unit for force. Fortunately all other constants drop out and it's simply I = v*p (where I=Intensity, v = velocity, p = pressure) and the units come out to be Watt per square meter
6. Intensity if a function of direction and position. For example a trumpet is significantly louder in front than in the rear. So the intensity is higher in front than in the rear.
7. To get the total power radiated by a sound source you need to measure the Intensity at points all around the sound source. Technically that's called a surface integral but it simply means: measure at a bunch of spots and sum it up weighted by the size of the spot
8. If you are "far enough" away from a sound source, there is a fixed relationship between pressure and velocity. p = rho*c*v, where rho is the density of air (roughly 1.2 kg per cubic meter), c is the speed of sound (roughly 344 m/s) so the Intensity becomes simply a function of the squared pressure and we can drop the pesky velocity for something like I = p*p/rho/c.
9. To make this more convenient the scientific community has chosen to declare the pressure of 20 micro Pascals t be the reference pressure as it's close to the human threshold (for people much younger than me) at 1 kHz. dbSPL is a logarithmic unit that uses 20 micro Pascals as a reference.