In the previous video, we had discussed sound pressure in its general form, with its SI
unit of Pascal. And we’d listed pressures ranging from the threshold of hearing, to
the threshold of pain, and a few common ones in between. Now, let’s try to plot this
on a graph, a one dimensional scale. Just to make it easier to visualize the scale,
let’s multiply all the numbers by some scalar value to make them whole numbers. A factor
of 10 power 5, or 10,000 should do us well. Once we scale up the values, we get a new
arbitrary range from 2 all the way up to 10 Million. Let’s plot this linearly. Meaning
that the difference between any 2 consecutive plot points is the same. Here, the difference
between the points is 1 million, and we have 10 points to make up 10 million. The plot
looks ridiculous. All resolution is lost at the lower range of values, where our ears
are most active. There is a lot of wasted real estate where we don’t really plot anything.
And if we do want to achieve a reasonable amount of resolution for the lower range of
values, we have to resort to decreasing the difference between consecutive points, thereby
increasing the length of the scale to something larger than this video window can hold, much
larger. And another problem with linear representation is that it doesn’t paint a true picture
about subjective loudness. For instance, the noise of traffic at a distance isn’t 100
times louder to perceive than a conversation with a friend for example. It’s definitely
a few times louder, but not a hundred. The reason why the linear scale is so intuitive
for us, is that everyday math revolves primarily around addition and subtraction. What we need
is to forcefully think about multiplication and division. Let’s reformat the scale,
and instead of the graduations being at every addition of 1 million, we can make it so that
the graduations are a multiple of 10 apart. More simply, we can represent them as powers
of 10. What we just did is create a logarithm scale. It’s a really useful scale for representing
exponential functions and exponential growth, where the range of values are massive. As
we can see, the logarithmic scale is pretty useful in representing our full range of values.
Let’s look at what a logarithm or log of a number really is. It’s just another way
of querying the power or the exponent function. Using the power function, we can state that
10 raised to the power of 4, is 10,000. Another way you can frame this statement is by stating
the log of 10,000 to the base 10, is 4. That’s all it is, the log function is just a rearrangement
of the exponent function. We can simplify this further by ignoring the base of 10, because
in most engineering disciplines, base 10 logarithms are the most commonly used variant, and it
is implied. In pure mathematics however, log to the base of e, which is the Euler constant,
is more commonly used, and it’s called the natural logarithm. You can differentiate them
in a calculator. log, is log to the base 10 and ln, is the natural logarithm, with its
base as e. Another simple way of looking at it, is by counting how many digits or zeros
a particular number has. log of 10000, is 4, since it has 4 zeros, log of 10 is 1, log
of 1 is zero and log of 25000 is some number between 4 and 5.
Mathematics out of the way, we can focus on how we want to represent the pressure scale.
We’ll ditch the scaled numbers and go back to the original table of data. A logarithmic
scale is honestly all you’d ever need. We solved the problem of lack of resolution with
the log scale. But we can go further. What we can do now, is instead of talking about
the absolute pressure value of some source, which isn’t particularly useful, we can
instead talk about how much more the pressure value is when compared to a baseline standard.
We need ratios. We want our scale to start from zero, because it’s nice that way, and
we need to set the baseline value, against which all other comparisons are made. What
value shall we assign to the baseline? Perhaps the threshold of human hearing? Alright, let’s
stick with that. So in our scale, any pressure value we want to represent will be compared
to the threshold of hearing of 20uPa, as a ratio like so. And to constrain the size of
the scale, we’ll take a log of this ratio. Beautiful, we’ve constrained all the pressure
values on this table on our new scale between 0 to around 7.
Turns out, something similar already exists. It’s called a Bel, and it’s a relative
unit of measurement used to express the ratio of one value of a power or field quantity
to another, on a logarithmic scale. You might wonder how it’s similar. Bel is infact a
unit first used in the measurement of electrical power in telephone applications. So it’s
not immediately apparent how it can be applied to the measurement of sound pressure. If you
look closer, it says power or field quantity. Pressure is infact a field quantity. Pressure
exists in a medium and can be measured at any cross sectional area of the medium and
is dependent on the position from the source. Meanwhile the source can be considered to
be releasing energy over time, this energy is what’s responsible for creating a change
in pressure in the medium. Consider this analogy. A heater in a room warms up the room. The
heater is rated to some electric power level, whereby it converts electrical energy into
heat over time. This causes the temperature in the room to go up. We can view this from
the lens of cause and effect. The heater or the electrical power is the cause, and the
change in temperature in the room is the effect. They are completely different units of measurement.
Electrical power is measured in Joules per second or in Watts. And temperature is measured
in Celcius. But there is a definite relationship between them, although the relationship can
be complex. The temperature that you feel depends on how well the room is insulated,
where the heater is positioned, and how far away you are from the heater. This exact analogy
can be applied to, let’s say a loudspeaker. The acoustical power of a loudspeaker can
be considered the source, measured in watts, and the pressure difference that it creates
is the effect, measured in Pascal, and a complex relationship exists between Sound power and
sound pressure which depends upon how reflective the room is, where the loudspeaker is positioned,
and how far away you are from it. We’ll go into a lot more detail into sound power,
sound intensity and why these are purely theoretical concepts, and we’ll explore this formula
in detail which outlines the relationship between sound power and sound pressure. But
for this video, if the measurements are done in the same setting, these properties can
be considered to be some constant value, and we can see that the power is directly proportional
to the square of the pressure. Let’s use this relationship in calculating the bel.
1 bel is said to be log of the ratio between two power quantities. If we substitute power
with the relationship expressing pressure we can see that the constant terms get divided
and cancelled, and we’re left with the square of the pressure. And by the properties of
logarithms, this can actually be expressed like this. And as we mentioned before we can
substitute the baseline pressure value to the threshold of human hearing at 20uPa. Most
of the time, we want a little more granularity and a slightly larger range of values, so
we use the decibel scale instead, where decibel is just one tenth of a Bel, and can be expressed
like this. This is it, this is the standard and accepted way of measuring sound pressure.
We can recalculate the values of the entire table, and this is what we get this, where
the decibel of sound pressure levels range from 0 decibels at the threshold of hearing
to around 130 decibels at the threshold of pain.
Here’s an interesting bit, which makes decibels a hairy unit to work with, and is a source
of confusion for a lot of people out there. When you are taking a ratio of the pressure
values, you are dividing one unit, by the same unit, which cancels the entire unit out.
And thus, the decibel is a dimensionless unit. Once it’s calculated, you’d have no way
to know what it’s a measure of. It could very well have been a measure of the sound
power levels, or the sound intensity. Or even voltage or electrical power, all of which
are valid ways of being expressed in decibels. For instance saying something like a sound
is 60 decibels loud, makes near to no sense, because it doesn’t give you any context
of what was initially being measured. To make it crystal clear that we are actually talking
about pressure, and that we’ve taken the reference pressure level as 20uPa, we append
SPL or Sound Pressure Level, after db. So stating something like a sound is 60 dBSPL,
makes perfect sense, since there is no ambiguity about what you are talking about.
That about completes our discussion on sound pressure level, but before we wrap up. I just
want to explore a couple of commonly encountered questions. Can a decibel pressure level be
negative? Absolutely. We’ve set the baseline or the reference value at 20uPa, the lowest
our ears can hear. But our measuring instruments can be tuned to pick up pressure values far
lower than that in anechoic chambers. If the ratio between the pressure values dips below
1, the log value turns negative. What’s the change in decibels when pressure
is doubled? If we doubled the pressure, what would that mean on the decibel scale? Let’s
find out. If a pressure value is twice the original, then the calculation boils down
to 20 log 2, which is around 6dBs. So every time pressure is doubled in Pascals, the decibel
scale increases correspondingly by 6dB. Similarly, if the pressure is halved, the decibel scale
reduces by 6dB.
Here’s the last question. Does doubling the pressure, double the perceived loudness?
Well, it’s subjective, as I’ve already pointed out before. Loudness depends on a
lot of factors like the frequency content of the sound, but a pressure increase anywhere
between 6 to 10dB would relate to a perceived doubling of the loudness.
In the next video we’ll take a look at acoustic power and intensity of sound, and how different
they are from sound pressure and why they’re relevant.