The lowest sound.
We'll talk about the measurement in a second
but first let's establish a particular threshold.
So the lowest sound energy that can be picked up by a human ear
is about 10 to the minus 12 watts per square meter.
So there's a few interesting things about this equation.
First of all, this I is how we talk about the intensity.
The intensity of the sound
which has to do with the energy of the sound wave.
We put a little zero here just to talk about the minimum energy.
So this is just a way of talking about the
minimum energy that we can pick up, the minimum intensity.
Another interesting thing is look how small the number is.
We can actually pick up very, very quiet noises
but one last interesting thing
and possibly the more confusing thing about this,
so we should talk about it, is this watts.
Why are we measuring the intensity coming from the sound source
not as how much power per square meter but in watts?
Which remember is a unit of power rather than energy.
So we're not talking about the energy per square meter,
we're talking about the power per square meter.
So why are we doing this?
Why are we talking about power which is again energy per unit time?
Well, the reason is this.
Suppose that I created a particular sound.
I banged really hard on a drum
and it created some sound energy and that sound energy started to move out.
There'd be a particular amount of energy there that you could hear.
But what if I spread out that energy in time
so that I let you hear parts of it very slowly
and somehow manage to spread out that same amount of energy
but over a long period of time?
You would expect that that sound would be much less loud
because I've spread it out and so the intensity of sound
does not depend exactly on the energy,
because again the energy can be spread out in time.
It depends instead on the energy per time,
how much energy is approaching you every second
and so that is a unit of power,
energy per time and so again we have the intensity here.
The smallest that the human can hear is this very tiny number
and again measured in watts power per square meter.
Going back to this measurement question.
I said that we have to have a sort of complicated way of measuring sound
because we have this discrepancy in how we perceive sound
versus the energy of sound and how the energy disperses away from the sound
and so this unit of energy is called a decibel and these decibels are,
as you can see here in this equation
10 times the log base 10 of the ratio of the intensity.
The total intensity of the sound you're hearing to the reference intensity.
Some reference in this case we could write it as the,
the lowest sound a human can hear.
So this equation might be a little intimidating
especially if we're not completely familiar with logarithms.
So let's go over these decibels a little bit
to make sure they make some sort of sense.
First of all,
the logarithm might be the most confusing thing in this equation.
This L-O-G, this log term.
You should definitely review logarithms
and make sure you understand what they are and what they're saying.
And so we're just going to do a very brief overview of what a logarithm is
and perhaps the easiest way to think about a logarithm
is to remember that a logarithm is an exponent, always.
So there's this term here,
this log base 10 of something, this term is an exponent.
But what exponent is it?
A log like this one is asking what exponent of 10
and that's why we have the base 10 there, it's log base 10.
I'm asking what exponent of 10 is required to bring that 10
to the value in the log which in this case is I over I zero?
And so that is what a logarithm is.
I'm asking you, what power do you need to raise 10 to,
in order to get to this number, I over I zero?
So for example, the log base 10 of 100 would be two
because I'm asking you what power do you need to raise the number 10 to,
to get to the number 100 and so the log of again 100 would be two
while the log of 1000 would be three and so on.
So you can see that a logarithm sort of compresses this line of numbers
because instead of going from 100 to 1000 to 10.000
we're just going from 2 to 3 to 4.
And so this sort of compresses our measurements
and allows us to measure much bigger numbers,
much more efficiency, efficiently.
The decibels, when we're talking about decibels here in sound,
is a way of measuring the units of this beta
which is in decibels and that's the reason we have the other 10.
This extra 10 put out in front of the log.
So we also have to multiply this number by 10 after we've taken the logarithm.
We have a few examples by this, of this.
After we multiply this number by 10,
we can ask ourselves what's the intensity
and what's the corresponding decibel for that intensity.
So you can see in this chart,
suppose we have 0.001 of the basic intensity that a human could hear.
The decibels would be minus 30
and the reason is the power of 10 we need to get to 0.001 is minus 3
and then we multiply by 10 so we have minus 30 instead.
We can go down this entire chart
so I definitely recommend you look at this
and make sure it has some sort of intuition behind it
as to how we arrive at a particular decimal number.
Again, so what we're going to do is multiply the intensity by 10.
So be aware that when we're multiplying the intensity by 10,
we're always adding a number of 10 to this beta, these decibels.
So in other words, look at the chart for the decibels as we go down.
When I multiply the intensity of the sound by 10
maybe going from the 0.1 of the original intensity
to just 1 times the original intensity.
We have added 10 to the number of decibels,
which we're calling beta here and so we go from minus 10 to zero in this case.
So this is a quick way to think about how to transition
from one unit of decibels to the next one.
And finally don't forget that this also goes in the other direction.
If I divide the power, the intensity of my signal by 10,
I will subtract 10 from my units for decibels and this is a good motivating,
motivating way to think about how we define decibels
and why we define them in the way that we do.
Finally, it's important to note that while I'm calling decibels,
beta in this expression,
decibels as a unit are called dB, a lower case d and an upper case B
and sometimes instead of calling decibels beta
as I've called in a variable here.
Sometimes people would simply write dB as decibels
and then this would just be a one way of referring to some number of decibels
aside from simply using them as the units.
So also, a few last things about the basic properties of sound waves.
As the sound is traveling outwards through the medium, in this case, air,
it can be losing some of this energy and losing this intensity.
So not only spreading out
and having to spread the energy across the surface area
but it could also lose some of the energy to the medium itself.
So you could have some of this energy lost to
may be friction in the medium and it could go to other places like heat,
heating up the particles as it travels through the medium.
This effect of losing energy from the sound as it's going out
is called damping or sometimes, attenuation.
So you could say that the amount of power or the amount of energy
is being dampened or attenuated
as your sound moves outwards through a medium.
So this sums up and summarizes and finishes our first discussion
of the initial properties,
the basic properties of sound that we should be familiar with.
What sound is, what's its physical basis is
as well as how to measure sound. And so next building on this,
we will talk about some more complicated ideas related to sound
and the dynamics of sound waves and that would be next time.
Thanks for listening.