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Video 3

Welcome to lecture 26 in the series of Acoustic Materials and Metamaterials. So, we have
already done 2 lectures on introduction to Acoustic Metamaterials. So, we know we have
got a brief understanding on what these materials are and what are they composed of? In
today’s lecture we will go through some historical development and study why the meta
materials came about and how they came about.
So, let us go through the lectures. So, the history of metamaterials it began with the
invention of the negative index materials. So, this acoustic metamaterials came first from,
the metamaterials discovered for, the domain of electromagnetic waves. So, what were
these negative index materials?
(Refer Slide Time: 01:12)

So, these were a type of materials that were used to manipulate electromagnetic waves and
these were those metamaterials, these are those metamaterials that exhibit negative values
of refractive index for electromagnetic waves at certain frequencies. So, usually most of
the classic medium it has a positive refractive index. So, in any in for example, in general
in optics, most of the lenses that you use etcetera it all has positive refractive index, but

certain materials could be made to have negative refractive index at certain desired
frequencies.
So, the here the refractive index would be a function of the frequency and at certain
frequencies the value of the refractive index will become negative. So, those materials
were called as the negative index materials. So, the people who are interested to know
more about this topic can; obviously, can go back in the literature for such and such
materials. And, they were used to manipulate electromagnetic waves.
And, then a similar concept was borrowed to build up materials which can manipulate
sound waves. So, just like acoustic metamaterials such kind of negative index materials
they also have a periodic arrangement of different unit cells.
(Refer Slide Time: 02:33)

And, in the same way the individual unit cells they are composed of conventional
materials. So, unit cells itself they have positive value of refractive index, within the
normal range as found in the traditional materials, but the combined effect of a collection
or arrangement of such unit cells will be, that it is able to bend the waves very sharply in
reverse direction or it attains a negative refractive index. So, this is the combined effect
becomes negative refractive index.

(Refer Slide Time: 03:06)

So, to show you to go briefly through the theory of this. Now, refractive index is defined
as ratio of the sine of angle of incidence and the sine of angle of refraction, when an
electromagnetic wave interacts with the boundary of the two different media. So, let us see
if we see this figure here. This is medium 1, 2 and sound interaction is taking place at the
boundary, this is θ1 is the angle of incidence θ2 isthe angle of refraction, then the refractive
index of medium 2 with respect to medium 1 will be:

n2/1 =
n2
n1
=
sin θ1
sin θ2

And most of the times so, this refractive index is usually a ratio or a comparison of medium
to 1 medium with another medium. And, most of the times there is an absolute value of
refractive index where, the medium 1 is taken as vacuum. So, when the medium 1 becomes
vacuum and the refractive index of vacuum is 1. So, if: n is equal to so, if:

n1 = 1 ⟹ n2 = n2/1 =
n2
n1
=
sin θ1
sin θ2

So, when vacuum when the first medium is taken as vacuum then the value of refractive
index that you get for medium becomes it is absolute refractive index.

(Refer Slide Time: 04:37)

Now, according to Snell’s law so, every interaction it follows the Snell’s law. So, by
Snell’s law:

sin θ1
sin θ2
=
c1
c2

Where, c is the velocity of the wave in the respective media. So, when the sound waves
they interact with any general medium in that case their refractive index again I am
restating. So, this is the overall expression:
n2
n1
=
sin θ1
sin θ2
=
c1
c2

So, this is the definition of refractive index and how it is related to the speed of sound in
the two different media. So, refractive index ratio is the reverse of the speed of sound ratio
in the two media. And, I have explained this concept to you again previously in the lecture
on acoustic metamaterials.

(Refer Slide Time: 05:35)

If you refer to the previous 2 lectures you can find this explanation there. So, the same
explanation goes here also. So, let us say interaction is taking place and both medium 1
and 2 they have positive refractive index. So, in that case this is the equation. So, this is
positive this is positive and this θ1 it varies from 0 to 90 degrees.
So, θ2 is limited and it can only vary within this domain, but if 1 of the refractive index
becomes negative for the second medium. So, suddenly the sound, the electromagnetic
wave is propagating it hits the boundary of a medium, which has a negative refractive
index.
So, in that case from this expression the θt or θ2, it will have a negative value. So, negative
value of sin means it will bend somewhere along this zone. So, this is taken as the positive
direction for θt

, this is taken as the positive direction for θi

. So, this is the convention that

has been being followed θi

this is the directions of positive the θi

, this is a direction for

negative θt

. Sorry positive θt

. So, θt negative means the will bend towards this. So, this

is the region of all the positive possible bendings that can take place.

(Refer Slide Time: 06:57)

In such a case sharp bending can be obtained and with appropriate choice sometimes you
can also have reverse bending. So, if this is of extremely high negative value, then this θt
can cross this particular region and it can go here. So, effectively what is happening is that,
clear your you are bombarding this material from a with the wave front in this direction,
but it is not able to enter the material it simply comes back.
So, if the appropriate choice of n2 is taken some negative n2 is taken then θt will be so
large, that it would not be able to enter the material, but simply bend around the material
and come back. So, that can be done and therefore, such materials they are heavily used to
make some super lenses where, very sharp bending of these electromagnetic wave such as
light waves can be obtained.

(Refer Slide Time: 07:49)

So, this figure shows the kind of sharp bending from a negative index materials. So, this
is a periodic this is a periodic layer of unit cells. So, these are the various unit cells here.
So, when the wavefront is incident from this direction here, then it is not able to enter the
material here it simply bends away and reverts its direction and goes back to the other
medium. So, very sharp bending is obtained ok.
(Refer Slide Time: 08:20)

Now, that we know that, what is the significance of having a negative refractive index. It
can help in manipulating the electromagnetic waves, in a very strong way not that

conventional materials cannot do. So, because this is not a course on electromagnetics or
the course of course, on acoustic optics. So, this that is why discussing about electric field
and magnetic field is not within a domain. So, I will directly give you what how this
particular concept can be applied to acoustic, how it came to be applied to the acoustic
domain.
So, from the electromagnetic theory based on Maxwell’s equation and Snell’s law of
refraction, there are two critical quantities or critical parameters of a medium, which is
called as the permittivity and the permeability which is ε and μ. So, these are the 2
important properties of a medium which define what will be it is refractive index. And, the
velocity is given by the velocity of the medium is given by:

c = (με)
−1/2

And, we know that the absolute value of refractive index will be what is it is the inverse
of the ratio of the speed of sounds.
So, absolute value is simply the n of the medium by the n in the vacuum, which is equal
to nm
n0
. So, this is absolute value n and will be c0
cm
, it will be the reverse of the velocity ratios

in the 2 media. So, this overall n can be written as:
n = √με

So, the refractive index can be written as root over of the relative permeability, because
vacuum:

√μ0ε0 = 1

And, that is why:

n = √με

From this expression here. So, now, we get is that this μ and ε are the 2 critical parameters,
this is the permeability and the permittivity that can control the refractive index of a
medium. And, now we want to obtain a negative index material. So, how do you obtain
this negative index material?

(Refer Slide Time: 10:50)

The way to achieve this is make this ε value and the μ value simultaneously negative. And,
therefore, such material sometimes are also called as double negative materials because
both μ, ε < 0.
(Refer Slide Time: 11:08)

So, if at certain frequencies both these values become negative what happens is. So, let us
say both permittivity and permeability become simultaneously negative at certain
frequencies, then this μ value is a negative value. So, it can be written as minus of its
positive value. So, this is some negative value which I am writing with a. So, let us say it

was 5, -5. This is minus and the absolute value of this which is 5 is also written as minus
into it is absolute value. And, the refractive index is:
n = √με

So, this can be written as this expression. So, if you separate these two. So, this becomes:

If: μ < 0 and ε < 0 ⟹ n = √−1 × −1√|μ||ε| = −√|μ||ε|

So, when both μ and ε are simultaneously made negative you get a negative value of
refractive index and that is why double negative materials came into existence.
(Refer Slide Time: 13:00)

So, the same philosophy is then applied to acoustic materials. So, let us give you an
analogy between the different parameters in electric in electromagnetics and acoustics. So,
just like we had μ and ε and we manipulated the value of μ and ε to get a negative
refractive index. In the same way for acoustic metamaterials, the critical parameters are B
and ρ which is the bulk modulus and the density, effective mass density.
So, when they are simultaneously made as negative, then c will be:

c = √
B
ρ
and c0 = √
B0
ρ0

So, anyways the refractive index will be the reverse of this velocity value, which can be
written as some relative quantity. So, it will be ρ this you can write as:


B/B0
ρ/ρ0

So, this becomes some density with this is like a relative density with respect to air.
So, some relative density with respect to air and some relative bulk modulus with respect
to air. So, this is how it can be written as the refractive index. So, when both ρ and this
value and this value are becoming negative. So, when this and this become negative. So,
the numerator will be negative, the denominator will be negative both of this ρeffective will
then become negative. Beffective will become negative, because B and ρ are negative, but B0
and ρ0 for the air are positive.
So, this is what we get? So, again using the same thing which we get is 343. So, we can
show some negative times of some number and some negative times of some number, you
can take out this factor. So, what you, so basically what I am trying to say is that in acoustic
materials the double negative does not exist experimentally, but theoretically it can be
made. So, usually refractive index can be changed or manipulated using this B and ρ
values.
And, we already and I have already discussed with you in the previous class what happens,
when ρ becomes negative and B is positive, and what happens when B becomes negative,
but ρ is positive. So, if either one of them becomes negative, then we get an imaginary
speed of sound and imaginary propagation vector.

(Refer Slide Time: 15:37)

And, how do we interpret these two quantities here. If, suppose some medium has negative
effective density, which means that the medium is expanding, when it is experiencing
some compressive force and the medium is contracting when it is experiencing some
tensile or tensile force. And, if: Beffective < 0 means it is a sort of a medium which
accelerates to the left, when it is being pushed to the right and vice versa.
Now, you have to take this into consideration that none of the naturally occurring materials
will actually have a negative ρeffective or a negative Beffective. So, none of the materials
which found traditionally they have a negative value of B and ρ, but when they are
arranged in the form of unit cells and they are arranged periodically the combined effect
comes out to be negative B or a negative ρ. So, let us so, this was the history of how the
concept of negative index material was applied to acoustic metamaterials. Now, based on
this some of the early acoustic metamaterials were proposed.

(Refer Slide Time: 16:53)

So, here you see here is that the concept of this metamaterial and especially acoustic
metamaterial was proposed by a scientist called Victor Georgievich Veselago in 1967.
And, it was proposed analytically so, there was just a theoretical concept that such kind of
materials can exist, but it is an irony that it was an idea much ahead of its time. So, almost
33 years later the first acoustic metamaterial was actually made and tested experimentally.
So, almost 33 years after its first proposal did we see an experimental verification?
(Refer Slide Time: 17:37)

So, it came in the year 2000 and the first acoustic metamaterial, which is fabricated was
called as the sonic crystals. And, we know that the meta, the acoustic metamaterials they
can either be of type of negative density, or they can be the type of negative bulk modulus,
or then there can be double negative materials. So, this sonic crystal which was proposed
was a better acoustic metamaterial with negative bulk modulus.
So, it worked on the principle of negative bulk modulus. So, you can see here the unit cell
is actually a hard led ball, which is covered by a rubber. So, here 1 centimeter diameter led
ball was used, which is covered by 2.5 millimeters layer of silicon rubber. And, they were
arranged in such kind of cube, they were put together in a cube and it was an 8 × 8 × 8
cube.
(Refer Slide Time: 18:39)

So, this was the kind of material proposed and the transmission coefficient. So, this was
experimentally made and then sound wave was impinged on it. And, the spectrum and the
other end was noted and the transmission loss was computed. So, what was the
transmission coefficient and this is not transmission and this is τ the value of τ. So, higher
the value, if higher so, a lower value of τ means less transmission which means higher
transmission loss.
So, as you can see here in this particular material, it does not follow the mass frequency
law, because according to mass frequency law, ideal mass-frequency law, if this is τ and
this is f, then as frequency increases the transmission should go down. Because, because;

obviously, the performance will improve. So, this should be like the curve or something
like that a uniform decline, but here as you can see, this the traditional materials they
follow this decline curve, but this particular material here.
So, this was what the when the individual balls they were their transmission coefficient
was noted. It was found to obey this traditional mass frequency law, but the combined
effect was this. So, although it did follow the mass frequency law, but there were 2 areas
where a major dip was observed.
So, as you can see even at a very low frequency suddenly you are getting a very high dip
compared to at high frequencies. So, what it means is that, even at this low frequency of
400 Hertz, the material can be can provide a very heavy transmission loss suddenly at a
low frequency. So, this was observed. So, this is the region where it showed exceptional
values.
(Refer Slide Time: 20:36)

So, the second form of the major acoustic metamaterial was then proposed in the year
2004. And, for all of this I have given you some sources which you can study as an
additional reading, if you are further interested to study about them, but I will cover the
sonic crystals, in our later lectures, in a detailed manner.
So, the second material was in 2004, it was proposed by Liu and Chan this was a double
negative acoustic metamaterial. So, it was theoretically proposed which will have a

negative bulk modulus and negative effective mass density, but so far experimental
verification is still not concrete.
(Refer Slide Time: 21:19)

Then finally, in 2008 the first complete model of a membrane type metamaterial came into
being. And, after this lecture we will begin with a discussion of this membrane type
metamaterial in detail. So, here the unit cell is what we have a small waveguide or a small
diameter ,sub wavelength diameter tube and in that you have some stretched membrane.
For example, you can have a stretched rubber attached on the top like this. So, this is the
stretched. So, here a stretched rubber was used, but any stretched elastic membrane can be
used and this is a section of a waveguide, or in other words this is a small section of a
heavy, but hollow tube. So, tube made of a heavy material, but hollow inside. So, this is a
hollow tube and an elastic rubber or an elastic membrane is attached on the top and this
particular metamaterial it works on the principle of negative effective mass density.

(Refer Slide Time: 22:36)

So, the transmission loss that is observed for this kind of material as you can see here is
that, this is again τ expressed in as percentage, to transmission expressed as percentage.
So, here also what you see is that at some low frequency region suddenly you observe a
very sharp dip, which is not expected, because at low frequency is the transmission loss is
going to be less for a traditional material, but a heavy dip is observed at these values.
So, this is where the material is breaking the mass frequency law, it is showing you
exceptional transmission loss or a very heavy noise control even at low frequency. So, this
is the kind of material that was used 6 millimeter diameter circular steel disk 300
milligrams of fixed mass at the centre of the membrane. So, this is a mass attached here.
So, this was the unit cell and it was arranged periodically in this series manner. So, a long
waveguide can be having this unit cell. And, some sound was impinged from this end and
it was noted in the other end and ta heavy transmission loss was seen. Similarly, in 2009 a
very famous experimental work was conducted by Zhang, Yin, and Fang.

(Refer Slide Time: 23:58)

All the sources are given here, you can study these papers separately. If you are interested
to do some further reading on how the what is the progress in the development of the
metamaterials in the last 20, 30 years. So, here what they demonstrated was the practical
application of bending due to a metamaterial.
So, what they designed was they designed a flat layer of material. So, usually a lens is of
this form, but this is not what they used they designed a flat layer of material, which was
able to focus the sound waves by achieving very sharp bending.
(Refer Slide Time: 24:38)

So, this was the kind of equipment that was used and here they demonstrated how this
particular metamaterial was could be used to focus 60 kilo Hertz of sound waves. So, this
shows a setup used and they shows a typical graph.
(Refer Slide Time: 24:51)

So, here this is a graph this is the graph of the focusing of the 60 kilo Hertz wave. So, as
you can see this is the wave is being focused here is only 60 kilo Hertz.
(Refer Slide Time: 25:11)

And this is the wavelength. Now, depending upon where the focusing so, what is the focal
length? So, depending on the frequency the focal length used to vary. So, different

frequency waves could be bent together and could be focused at a point at different
distances. So, this shows the variation of the focal length with the frequency. So, what it
means is that a 60. So, this is in kilo Hertz. So, which means that a 60 kilo Hertz wave will
be focused at a distance of 60.60, let us say about approximately 62 millimeters away from
the layer of material.
Similarly, it will be able to focus a 64 kilo Hertz wave at a distance of so, all this is just
so, at a distance of approximately 45 or 46 millimeters away from the layer of material.
So, the kind of lens can be designed which can be used to bend the sound waves, bend the
sound waves, very sharply and focus them at a particular point. So, with this I would like
to end this theoretical lecture on the various progression and the history historical
development of acoustic metamaterials.
Thank you.