This paper will comprise a discussion of quantifying loudness and will make a connection between the amplitude of a sound wave (sound pressure) and the qualitative notion of loudness. It is presumed that the reader has an intuitive understanding and experience of sound at a particular point (in a room, on a city street, or near an airport listening to jets taking off) without worrying about how the sound is produced. Connecting loudness to the source of sound energy is important, because as we know from our direct experience, how loud we perceive a produced sound depends on the distance between the source and our ear. Today, we'll quantify how loudness depends on the amount of sound energy produced and the distance between the source and our ear
Where does the sound energy come from? The answer is that some object has converted energy of a different kind into sound energy. Typically, there is some sort of mechanical motion, a vibration of some sort, so mechanical energy is converted into sound energy. How loud a sound is depends on how rapidly the object converts energy into sound energy. So the relevant physical quantity to relate to loudness is the sound power produced by the source, or the rate at which sound energy is produced versus time. Referring back to earlier knowledge, power is the time rate of change of energy and has units of Joules per second. We give a new name to this unit: 1 Watt = 1 Joule/second. The Watt is the unit of power.
We will introduce a logarithmic scale for sound power, just like we did for sound pressure, because our perception of sound loudness is logarithmic. We define the sound power level from a source as
sound power level = LW = 10 log( W / W0)As with the defintion of sound pressure level, we are forced to first find the ratio of the sound power to some reference power level. In this case, the reference sound power we'll use is W0 = 10-12 W. The "units" of sound power level are in decibels. This is really a dimensionless quantity. We quote the sound power level in decibels to remind ourselves that its a logarithmic quantity with a specific reference power.
The geometrical shape that has all points equal distance away from a source is a sphere. The loudness we hear depends on the ratio of the area of our sound collector to the total area of the sphere surrounding the sound source. This motivates the introduction of another physical quantity associated with sound waves: intensity. The intensity of a sound wave is the amount of power in the wave per unit area and has units of W/m2. The intensity of a sound wave depends on how far we are from a source. If we label that distance as R, then the sound intensity is
sound intensity = sound power / (4 pi R2)This says that on the surface of a sphere centered on the sound source, all points get equal intensity which agrees with our intuition. The sound intensity is just the ratio of the sound power to the surface area of a sphere of radius R (surface area of sphere = 4 pi R2), where R is the distance we are from the sound source.
We now have something that relates to our perception of loudness. This means that we should introduce a logarithmic scale, once again. We talk about the sound intensity level
sound intensity level = LI = 10 log ( I / I0 )with the reference intensity defined as I0 = 10-12 W/m2.
Both sound pressure level and sound intensity level are purported to measure the same thing: the loudness of a sound that we hear. This means that they better be equal. They are for the following reason. The sound pressure level can be written as
sound pressure level = Lp = 20 log ( p / p0 ) = 10 log ( p2 / p02)The last step in the chain of equal signs makes use of a property of logarithms, log( An ) = n log A. In the above equation, n = 2 and A = p / p0. The next step to show the equality of sound pressure level and sound intensity level is to realize that the intensity in a sound wave is proportional to the square of the amplitude of the sound wave. This means that p2 / p02 = I / I0. The end result is that the sound intensity level and the sound pressure levels are identical quantities!