. Ah Good morning everybody . So, today we will
going to start the lecture number 4 and the module number 1. in the first week module
. So, today we will going to discuss about the sound pressure and the intensity level.
If you remember in the last two lecture what we have discuss is that is the propagation
of the sound, we also discuss the frequency and the octave band.
So, now, let us proceed further and let us know what is the sound pressure and how it
can be measure and how that particular sound pressure and the sound intensity can be
mathematically converted to the sound levels. So, in this lecture we have again two objective
as a learning objective. So, what you will want to do in this particular half an hour
lecture is that, we need to convert or we need to know how to convert the intensity
and the pressure to a particular sound level and also we will like to know that how
can this particular sound levels of from the multiple sources can be added and what will
be the result and sound level for that the for the additional all.
So, here we need some kind of operation, some kind of a mathematical operation to know
about. And this mathematical operations are well known to you, but that for the the
the starting point of view, let us brush up those mathematical operations and we need
here some logarithmic operation . In the logarithmic operation the first operation
is something like this, which is also you know that is log a if it is x, then 10 to
the power x it is equal to a. The next operation what we also need is that if though 2 log
terms are added. Suppose log a plus log b, it will be log a
b. The third operation what we need is that is the subtraction from one to another log;
that means, log a minus log b will be reciprocal that is the log a by will b log a by
b; and the third or the fourth one and the last one what we need is that if log a to
the power n is always equal to n into log a . So, this four basic operation of log will
be required to understand the basic principles of the sound pressure level and the sound
intensity level . Now, let us know how we actually here human hearing.
So, what we see in this particular board is that, the cross section of human ear. You
must have seen this particular cross section of the human ear in your school days in your
biology book . So, here as we understand from the first lecture is the sound is a wave
transmission, and that wave transmission gives in the ear in a multiple layer, which
will some layer will be compressed and the the next layer will be radar fact or the
extend and it will move from one point to the other .
So, when it moves from one point to the other, the wave propagation towards the if it is
towards the ear it will enter into our ear and how it will actually sense the brain,
let us discuss that . Suppose this is the ear cross section, the
similar ear cross section and it has some some some part. So, these four parts are may
be the first three parts are called as a the the the outer ear part, which is has a pinna
which is like this and then there is a auditory canal over there and then auditory canal is
actually terminated to the eardrum. And if there is a particular propagation and if
I am I am showing it in a flow chart diagram, the external ear that is from here, the pressure
which followed by a follow the path of the auditory canal and it will strike to the the
eardrum. And the total pressure of the particular channel or the canal auditory canal is equal
to the atmospheric pressure plus the sound pressure.
The P atmospheric plus P s and it will strikes to the ear drum. And there is a another cannel
or another tube that is there in through our mouth to the the to the cannel. So, that is
called as a the eustachian tube and this eustachian tube will only take care of the atmospheric
pressure, and it will also proceed to the ear drum. If you see I have animated a
bit, this particular this the the the the red arrows or the pressure from the air pressure
that is the atmospheric pressure and the the sound pressure and another the yellow arrows
from the eeustachian is the only atmospheric pressure.
So, the ear drum basically under two different pressure scenario, one is P atmospheric and
plus P sound and from other side it is P atmospheric. So, finally, the ear drum will vibrate because
of the the resultant pressure of that, that is two the P sound pressure that
is the the two mutually perpendicular and and opposite directional not
perpendicular, mutually in a in a in a linear and a opposite indirection. So, the
sound pressure, will the P s will actually vibrate the eardrum. What happened in the
next operation is that. So, this ear drum is connected by the three
is middle ears very small bones and there are three bones malleus incus and stapes is
linked one after another, and that is actually transmitting the the transmitting
the particular the three ear bones to the cochlea.
And cochlea is full of some liquid or so. So what happened is that, when this ear drum
is vibrate by the sound pressure it will going to vibrate the middle ear bones this three
bones and there is a micro amplification, micromechanical amplification I must say,
because this is a three bones are a hinge together and it will be the amplify the thing
and it will be actually impulse has been provided to the cochlea. And this three are the bones
and the cochlea are called the the in interior part of the ear and then finally, cochlea
sense that, cochlea sense that and then it is the behind the cochlea there are auditory
nerves. So, there the cochlea sense that particular
vibration from the three middle ear bone and it provide the some mechanico electrico kind
of a conversion, the mechanical to the electrical conversion and this electrical sensor pulses
through the nerve it will sense in our bend .
So, this is at a go, we can actually again see that particular flow chart. So, this
is the ear drum vibrate by P s that is the pressure from the sound, sound pressure than
middle ear vibrate micromechanical amplification gives to the cochlea, from the cochlea a mechanico
electrico conversion and then it is going through the the auditory nerve to the
brain . Now next is let us see the how the brain
and our sensation of the sound is perceive by the brain. A psychologist weber suggested
this particular phenomena or this particular proposition, that is the subjective response
is the function of change of stimulate is proportional. So, if it is proportional. So,
if it is something like a the change of responses R, this response may be the weight,
this response may be the light, there is a slight change in the light whether I can identify
that particular change or not, if there is a weight in my hand I just increase a bit
of extra weight can I sense that one that is the stimuli the change of the stimuli and
the R is the sense. So, mathematically it can be expressed as
the the change of R is equal to change of stimuli by the act the the initial stimuli
that is the cap S. If I integrate then this two side, then I will get a mathematical expression
like the R is equal to log of S and there is a integration constant are C and then further,
this integration constant constant C can be clubbed into this logarithmic part also, and
then we can write that R is equal to k times log of S. So, this is a particular way the
human being sense for weight for any scenario like a light or may be temperature or maybe
the sound. And we see that in the sound in in in case
of a the auditory case auditory sensation, we can sense the very minute change not in
a specific way, but if there is a change in a bigger order then it may be sense in a better
way. And this particular change is very much adopted or very much sensible if the frequency
or the level of the sound is in the lower order. This particular change the same change
if it is in a higher order, we may not sense that one.
So, that is why we can say finally, that a human being or a we here logarithmic. So,
the expression of the sensation that is the sound pressure or the sound intensity finally,
translated to a logarithmic scale and that will give us the sensation, what we have just
now discussed or the see from this particular webers proposition .
So, let us take a graph and in the the the y axis let us have the sound pressure
starting from 2 to 2 into 10 to the power minus 5 mega pascal 220. So, it is a huge
range huge range. So, almost 10 to the power 7 a 6times of the the the sound pressure
changes and in a another side let us write down the sound intensity and as you know the
intensity is the watt per meter square or the power per particular square print of
the the foot print of the area. So, they are also it is equivalent to 10 to
the power minus 12 watt per meter square two one watt per meter square. It is almost about
10 to the power 12 times higher from the lower to the higher level. And if the sensation
or the sound pressure which is actually followed and to enter to our the enter to our ear is
at around 2 into 10 to the power minus 5 newton per mm square or in the intensity terms 10
to the power minus 12 watt per meter square. We start hearing that is the minimum amount
of the the intensity level or the minimum amount of pressure sound pressure is is needed
to the the human being to hear some sound and that is why it is called the threshold
of audibility . So, if the sound pressure level is below that, we we cannot response
that the the to that particular sound. And if it is higher than 2 to the power 10 to
2 into 10 to the power minus 5 yes we hear better, and further better if the intensity
and the sound pressure is further up , but there is the limit.
If the particular pressure is 20 newton per mm square or the intensity is one watt per
meter square, then it is start paining to our eye paining to our ear and that is
very very comfortable, and that particular point or that particular intensity is called
the threshold of pain it is the discount foster. Sometimes it is also referred as the threshold
of this discomfort and further it may go further, it may go further 10 to 12 12 to
15 percent further, but it will start giving lot of problem to our ear and finally, the
ear drum may rapture and that is going to be a serious concern.
So, we will try to limit our the hearing pressure level, this 2 into 10 to the power
minus 5 to 2 20 and the intensity level 10 to the power minus 12 to 1 watt per meter
square. Now this are the some physical factor or the physical parameter, we have to translate
this physical parameter to a sound pressure level or the sound intensity level.
So, let us see the what is the sound intensity level . So, this graph and in this graph what
happened is that, the sound intensity level which is called as a SIL is defined as logarithmic
of I by I reference and if it is so, it is called bell and if it is 10 into logarithmic
of I by I reference its called disable. Now what is I and what is I reference let me see.
The I reference is the minimum level of the the pressure or the intensity, that we
required for the threshold of audibility . So, this is that particular 10 to the power minus
12 watt per meter square and what is I? The actual intensity of that of the sound. So,
suppose I am going to calculate some intensity which is in between 1 and 10 to the power
minus 12. So, that suppose 10 to the power minus 5.
So, the I value of I will be 10 to the power minus 5 also . So, here from here let us see
if the if the if the the sound pressure is 10 to the power minus 12 . So, then it is
one this logarithmic value is one the log 1 is 0. So, the it is 0 decibel almost no
sound just starting of the audibility. And if it is the higher suppose it is 1.
So, I put one over 10 to the power minus 12 over here, I may say take the logarithmic
and multiply with 10 this is 120 disable . So, it is painful its threshold of pain is 120
almost we may go up 240 also and this is very much painful and very much not not that much
that the hygienic and the peak going to rapture the the ear drum or so .
Very similarly we can find out the sound power level also. Sound power level is also define
in a same way in a decibel scale log of W by W reference into 10. And this w reference
is 10 to the power minus 12 watt because power the the the unit of power is watt, and it
goes up to 10 the 1 watt or so, where there is a and if you convert the same way it is
0 and 120. Now, let us see the sound pressure level which
we have just discussed in the the first graph. The sound pressure is again fluctuating from
2 into 10 to the power minus 10 to 20, and sound pressure level is defined as not 10,
but 20 into logarithmic of P by P reference as the intensity and pressure is related with
the proportionally is pressure square the intensity is proportional to the square of
the pressure. So, how it is we will discuss in the fifth
lecture and how it is actually the the proportional to the pressure square. So, here also the
similar manner we can find out the our P reference is that the minimum, which is the threshold
of audibility which is 2 into 10 to the power minus 5, the P the whatever may be the value
of the pressure that I will put over here, and if it is 20 that is the highest level
if I put 20 by 2 into 10 to the power minus 10 log and multiply by 20 its gives me 120
disable . So, by virtue of this if I go back and then we have derived the intensity
level, we have derived the power level, we have derived the sound pressure level and
all are in decibels . So, the y decibel y decibel because the we
we everybody knows the Alexander Graham Bell on the honor to his name honor to
his scientific development in our the the world community we have or the the scientific
community has proposed the the unit of the the sound pressure level as bell or decibel.
And in the left hand side you will see that from instrument, which is the sound level
meter and the sound level meter is used for the measuring the sound level that is in decibel
or so. So I have a sound level meter with me and
if you see this sound level meter it has a power button. So, you have to switch on the
power button and then there is a the range you can said the range, we can said
the range from the minimum to the medium to the maximum and there is a microphone attach
to this particular the the the arm on the top and as you see I am speaking. So, this
is going to be be some kind of the the sound pressure level.
Now, it is 65 70 68 68 62 like that . So, . So, some sound it will catch and then
it will going to give you the sound pressure level directly .
So, the sound intensity to the sound level conversion. So, now, let us play with those
formula and see how the sound intensity and the sound pressure and the sound pressure
to the intensity or those levels can be taken into consideration. So, here in this slide
we will discuss the sound intensity . So, the sound level conversion. So, as we know
this is the formula the level L is equal to 10 into the logarithmic of I by I reference.
So, I have a small small very small problem that suppose the intensity is 0.005 watt per
meter square what would be the intensity. So, as you put it everything in the value,
everything in this particular every value in the particular formula and the get the
ratio and multiply with the 10 log and multiply with the 10 and you get 97 dB which is pretty
high . Then the the reverse order the sound level
to sound intensity conversion. So, the sound level to sound intensity conversion
you have the same formula and from the formula, you derive like the then the log of I by I
reference will be [laughter] by 10, because you see the if you just go with this formula
and then if it is I by I reference will be 10 to the power L by 10 and finally, I am
interested to find out the I. So, the I will be equal to I reference into 10 to the power
L by 10. I reference we all know that is 10 to the
power minus 12 and l is the sound level that is d|B how much is the decibel that you can
put and finally, we can find out what is the value of i. So, suppose a a SIL of the sound
intensity level is 65. So, I will put 65 over here. So, finally, it is 10 to the power
6.5 and the I reference is your 10 to the power minus 12. So, finally, this is the watt
per meter square that is the the intensity .
So, I have listed down some intensity or the sound level in fact, in the dB just
to give you a kind of a idea that what is the dB that exist in our day to day life.
You sweeping of the dry lips is very very low level of or the sound which is around
10 15 or so, background noise in the television studio where there is kind of a broadcasting
or may be telecasting is of of a happening over there, we required very low amount of
sound level which is prescribed as 20 decibel. Library, very quit area, bedroom very quit
area 30 to 40 is recommended, residential zone recommended as 50, now normal conversation
which we are doing now it is more or less 60 we have seen in this sound level meter
60 to 70 it is fluctuating. Vacuum cleaner, vacuum cleaner is a sound produce producing
the equipment which is if you in front of one meter of that particular vacuum
cleaner gives you around 70. Heavy city traffic is 80; the pneumatic drill is very bothering
sometimes it is the 90 to 100. The discotheque it is the noise area very noise area in fact,
and its almost touching to the the threshold of pain sometimes it is 100 to 110 and if
it you if it is very near to jet aircraft, around 100 meter net to jet aircraft it is
really painful 140 . Now let us see the the how the addition of
sound levels can be done. So, I actually have to encounter or in a in a in in a any kind
of a sound physicist has to encounter different sound level multiple sound level. So, here
we will discuss the how this two sound level can be added together and in a logarithmic
way. So, sound level 1 is having a level 1 L 1 dB. So, that can be translated to the
I 1 as as you know there I reference by 10 to the power L 1 by 10, and there is a sound
level 21 or sound source 2, which is available L 2 and that is also give you a I mean very
similarly I 2 equal to I reference into 10 to the power L by 10. L 2 by 10 and
this sound intensity can be added arithmetically, not the sound levels that is the decibel cannot
be added the arithmetically . So, suppose there is a 60 decibel and there
is a 40 decibel you cannot add and say it is a 100 decibel. You have to actually
transferred the 60 decibel to the intensity its the the respective intensity and the 42
its respective and then you add the the the intensity arithmetically and convert that
intensity to the decibel. So, here also I have to now add this arithmetically
I 1 and I 2 and then this I 1 and I 2 gives me the a new I total the new intensity, and
then I will put it into that and finally, the value will be like L the total intensity
is this and the total level now you will be or the final level will be now will be
the this comes from the first part intensity 1 and this is comes from the second part intensity
2 and if you provide in the I total by I reference you will get the the total
intensity . Now addition of the sound level again let
us further say that if this two levels L 1 and L 2 which we have just now discussed is
similar. So, there are two sound source give the similar output. So, L 1 is equal to L
2 and that is equal to L. So, definitely that the intensity of I 1 and I 2 also will be
similar. So, the total intensity of the 2 y twice of I and finally, it will give instead
of in in place of the total intensity is now 2 I by I reference now we can separate it
out, one is the 10 log of I by I reference plus 10 log of 2 . So, what is this? What
is this 10 log of I by I reference it is nothing, but the anyone of the level .
So, this is the level which I am going to add plus 10 log 2 is log 2 is 0.3m and 10
log 2 is 3. So, if two sound source of equal in nature equal output equal decibel level
is added, suppose 30 and 30 is added. So, the resultant will be 30 plus 3 33. So, that
its what tells about . So, next we will see the sound the the
addition of the sound, addition of the sound also can be infer from this particular graph,
but how to read the graph. This in the x axis it is the difference between the sound level
is given and the y axis the addition will the at to the sound sources it is also
given . So, let us go to the a small problem, where
I am adding a 60 decibel sound to the 65. So, by virtue of this particular expression,
I got 66.2. Another example I get 90 and 95 and I express that in the mathematical form
and I am getting 96.2 I am getting taking the help of the previous equation. Now what
I see is that in the both the case it is a similarity what is this similarity? 62.2 66.2
is nothing, but 65 plus 1.2 what is 65? 65 is the out of this is the higher one 60 and
65 the higher one is 65. So, 65 plus 1.2 and what is this? six 96.2
is 95 plus 1.2. 95 is what the higher of this two. So, if the difference between the two
level of sound is 5 dB add 1.2 dB for the largest sound that is actually translated
over here. Suppose the difference between the two level
of sound is 5 dB of my I mean previous example, you go from the x axis to here and then you
go here. So, add 1.2 one point this is one, this is 1.25. So, 1.2 to the larger sound
. So, if there are two sound which are equal in nature. So, there is no difference. So,
the higher and lower there is no difference. So, difference is 0.
So, you have to add sorry you have to add 3 dB 3dB with that that we have already discussed
in the
previous to previous slide. And if there are n number
of sound sources suppose it is not to two there are more more more L 1 2 3 4 more, then
we can just rewrite this particular equation in a this way and finally, which is summation
sin, we can find out the what is the total logarithmic or total logarithmic and finally,
multiply by the that into 10 to the the L total or the level total level .
So, at the end of this this particular lecture number 4, let us have some homework
for you. The first one is that the the intensity of
the sound if it is 0.004 watt per meter square and suppose another sound is of the sound
pressure of 1.2 mega pascal, can you can you find out the related sound levels. These
two are not equal, but we have to find out for one also, we have to find out for the
second one also . So, just go to the previous slides find out the how to convert and while
converting in have to remember one thing, that is when it is intensity and when it is
power. So, it is 10 log and when it is pressure it
is 20 log and it is the the intensity, which I am going to take for finding the level divided
by the the the threshold that is for pressure it is 2 into 10 to the power minus 5, for
intensity it is or the this thing it is 10 to the minus 12 .
Ah There is a second homework also, suppose I have two different sound levels from two
different source one is 58 dB and one is 62 dB . So, if they are added. So, what could
be the resultant sound level? So, you have to do the this thing by mathematical computation
with the mathematical the formula, do it by mathematically and then check your result
from the graph, that the graph we have discussed from the graph and finally, match
your result. There are some bibliography, there are
some books has to be referred for this particular lecture and those are the books you can
find it in the library or maybe we can go to the some library and get some books
are available in E format also. So, thank you for joining in this lecture number
4, in the next lecture we will go to the the that will be thus the last part of our
acoustical physics and we will discuss something and on the the near and far field propagation
and also the loudness. Thank you .