Greetings, the subject is vast and in today's
class we will be concluding the second unit
on Oscillators, Resonances and Waves. One
can discuss very many very fascinating aspects
of resonance quality factor and so on and
so forth.
Our plan in this course is not to give a very
exhaustive review of these topics, but to
introduce and graduate students to basic principles
of oscillatory motion and damping, and other
things that we have been talking about to
get some sort of an introduction. We will
provide some further acquaintance with important
consequences of oscillatory motion leading
to the behavior of resonances and waves.
So, here is a picture and you can actually
see a movie. There is a website at the bottom
here and you do not have to write down this
link but, if you just Google this Tacoma Bridge,
you will get this link easily on the internet.
You can actually see a video; this is Tacoma
Bridge in Washington State. This was a bridge
about nearly 2 kilometers long, one of the
largest suspended bridges of that time. This
was in 1940 but, this is a very famous story.
What happened is that this bridge collapsed
in a very dramatic way on the 7th of November
1940.
Actually it turns out that it was for reason
that this happened because of the strong winds
which were blowing. These winds generated
certain oscillations in the bridge; the bridge
had its own internal natural frequency of
oscillation. The wind speed at 50 to 70 kilometers
per hour they produced a driving force; a
periodic driving force because of the manner
in which the winds were blowing and it led
to a resonant type of phenomenon which eventually
causes the amplitude of oscillation to become
so large that the bridge could not sustain
itself and it actually snapped.
So, this is the movie that you can see and
the basic phenomenon. It is not nice to think
about tragic events in terms of mere physics
and differential equations but, essentially
that is precisely what it really voice down
to. The important thing over here is to understand
that the differential equation that we set
up in this is not just abstract mathematics;
it is a real, it describes real physical phenomenon
and then it is real events which can be analyzed
and understood in terms of the differential
equations and the damping coefficients and
so on.
This has got consequences not just in classical
mechanics but, also in quantum theory and
here is a situation in which I do not want
to find anyone of you because I hope that
you will not need it.
This is the picture of MRI scanner and this
is a very powerful tool, diagnostic tool in
modern medical technology. This is based on
the principle of nuclear magnetic resonance,
so this is again a resonant phenomenon. This
is the quantum phenomenon in which, you have
got an intrinsic property which a nucleus
has called as the spin and I will of course,
not discuss that over here but, it is this
spin which when subjected to a certain periodic
electromagnetic oscillation that leads to
a resonant phenomenon.
Then when you explode the properties of the
magnetic resonance, you can use it to map
individual biological trends in the body at
a cellular level and that because of very
powerful diagnostic tool in medical technology.
So, the phenomenon of resonance will come
in classical mechanics, quantum mechanics
or I do not know 100s 1000s many examples
and it is for this reason that this is an
extremely important subject to learn.
Furthermore, the subject of oscillations is
a fundamental importance in understanding
how information is transmitted from one part
to another, how propagation of energy takes
place from one part to another. Digital signal
processing, for example and we will get some
kind of acquaintance in this last class to
some of these ideas; what is the wave phenomenon
and how is the oscillatory behavior at the
bottom of all of this analysis? So, we will
first look at a pulse and you know what a
pulse is; you just feel at a wrist, you can
feel your own pulse and you see periodic pulsations
which are generated by the bio rhythms at
a certain regular frequency with which the
heart is pumping blood through your body.
So, what is crossing a particular point is
a pulse and this pulse is sometimes also called
as a wave packet. It is actually made up of
a large number of sinusoidal waves and these
sinusoidal waves they come from the solution
to the oscillatory problem that we have discussed
earlier. So let us see, what a pulse is really
like; so here you see in this picture a pulse
at time t equal to 0 and it has got a peak
which is centered at x equal to 0.
You have got an x axis and the x equal to
0 over here under this vertical black line
and this is where the peak of the pulses and
then this pulse which is a superposition of
a large number of sinusoidal waves. These
different sinusoidal waves each has its own
wavelength which is lambda, each component
wave travels at its own speed which is called
as a phase velocity. This phase velocity is
equal to the product of the natural frequency
nu times the wavelength.
At a later time, at t equal to greater than
0 the pulse will be at a different location.
It will move away from the origin through
a distance vt depending on the time at which
you are seeing it. It will be described by
since it is moving with time, the argument
will be phase shifted through the factor vt.
Now, the shape of the pulse in the example
that I have considered, has not changed. This
is going to happen when each component you
know, all the components travel together.
This happens in a medium which is said to
be non-dispersive. So, each component you
know many different component which must be
superposed to generate a wave packet or a
pulse when they travel together and they can
travel together only in a medium which is
said to be non-dispersive.
So this also gives us the definition of what
is a dispersive medium and what is a non-dispersive
medium. The shape of the pulse must remain
intact as time progresses. Now, lot of this
analysis is due to Jean Fourier and he lived
between 1768 and 1830. He was at the Ecole
polytechnic at Paris and one of the very brilliant
mathematician and physicist who contributed
so much to the analysis of wave motion.
We find this being used in just about every
branch of physics in engineering with including
quantum mechanics and optics electro dynamics,
you name it and we make use of Fourier methods.
So, will get some idea about what this analysis
is about. The Fourier theorem can be stated
in some more simple terms and I am not going
into the rigorous mathematics of this because
we really do not have the time for it.
But essentially in some sense, what it does
is tells us that any periodic function, any
phenomenon which occurs with a certain periodicity.
It can be anything, which is a repetitive
phenomenon which repeats itself at a certain
frequency. It can be written as a sum of simple
oscillating functions namely the sine and
the cosine functions; now this is an amazing
feature.
This is absolutely incredible; that you look
at any periodic function and you can always
write it as a superposition of the sine and
cosine functions which are very simple functions
to use. Everybody knows what a sine and cosine
function is and in terms of the sin and cosine
functions, you can write anything and everything
that has got any kind of periodicity.
This comes from the Fourier methodology and
we will see how these methods work. Let us
plot a function which I have written on this
equation for f x this function is defined
in terms of another function H, which is called
as a Heaviside function . So, H of x is a
Heaviside function, it is also sometimes called
as the step function and the definition of
the Heaviside function is very simple one.
Let us first define the Heaviside function.
Once, we know what the Heaviside function
is then instead of the argument x, we must
put the argument x over L and then we get
the first term; subtract from it the Heaviside
function for a different argument which is
not x over L but, x over L minus one. Then
you subtract from this result after multiplication
by 2 a factor of 1 and you get the function
f of x; so that is how you will get the function
of x.
So, everything is defined in terms of the
Heaviside function and the definition of the
Heaviside function is very simple. It is equal
to 0, if x is less than 0 and if it is greater
than 0, it is equal to 1. So, that is part
of the reason it is called as a step function
because at x equal to 0 its value suddenly
in one step shoots up from 0 to 1. We are
interested in plotting the function f of x
in the range 0 to twice L, where L is some
length parameter.
Let us first have a look at the Heaviside
function. So for x less than 0, it is 0, so
it is indicated by this green line and for
x greater than 0 it is equal to 1; so that
is what generates this step, so this is the
Heaviside step function . Now that we know
what H of x is, we have to find how to plot
H of x over L in a range of x which goes from
0 to 2L. That is the fairly straightforward
thing to do and I will let you work it out
in details in your notebooks. I will plot
the result over here but, now you know how
to do that.
So, you have an x axis, this is the 0 of the
x axis, here is a point at x equal to L and
here is a point at x equal to 2L and it is
in this range that we have to find what this
function f of x is. If you just substitute
the value of the argument to x over L and
then construct this difference multiplied
by 2, subtract 1 out of it you will get the
function f of x. Now this is what the function
f of x turns out to be, you see that it is
a square wave. So, it is coming from a sequence
of these step functions the manner in which
you have defined the function f of x and this
is actually a square wave.
Now suppose, this square wave wants to repeat
itself, then you have a phenomenon which is
periodic. You would expect that this can be
written as a sum of sine or cosine waves,
now this does not quite look like a sine function.
Everybody has a picture of a sine function
in your mind, it has some similarity to a
sine function but, the difference is more
visible than the similarities. Because this
has got sharp edges right, a sine function
does not have any cuts, there are no angularities
in the sine function. It is very smooth function
which varies between minus 1 and 1.
So, here also we have chosen a function which
varies between minus 1 and 1; it is a repetitive
function. It does not have to be minus 1 and
1; it can be from minus n to plus n you can
always scale it by an appropriate factor and
normalize it to unit magnitude, so that is
not an issue.
Now, it turns out that this can actually be
written as a sum of sine waves and I strongly
encourage you to do this exercise on your
own using some graphical plotting routine
on your personal computers. I have used graphics
software which comes in the software called
region to plot this graph and this function,
this is the function of theta which I have
generated here which is written as a superposition
of the sine function, but it is a very peculiar
superposition. The argument is n theta where,
n can take only odd numbers all the way from
1 through infinity but, it will take only
odd numbers 1, 3, 5, 7, 9 all the even numbers
are omitted.
Then each sine term is divided by the corresponding
n, so it is 1 over n sine n theta summed over
n going from 1 through infinity. This is just
some normalizing factors, so do not worry
too much about it; if you do it, you can get
depending on how many terms you put in this
summation, this summation is go up to infinity
and if you take it too seriously, you will
spend your life time and beyond doing this
summation. Not just your life time but, also
beyond that; so I do not want you to spend
the rest of your life time to doing it.
So, just take the first term, n equal to 1;
n equal to 1 is sine theta divided by 1 which
is again sine theta and then you get this
black curve, this one . This is sine theta,
here it goes and this is the usual sine theta
that you see; then, you add to that sine 3
theta divided by 3; then you add to that sine
5 theta divided by 5 and if you go up to 5
theta and just stop there. You are supposed
to go up to infinity but, you decided that
you will not do it. So, you stop just at two
terms, forget the infinity.
At 5 theta if you stop, you get the red curve
, this is the red curve which goes like this,
it vigus over here, vigus further and then
it dips to negative values, vigus over here
and gets back; if you go further up to 9 theta,
if you add to the previous sum the 7 theta
by 7 and then also sine 9 theta by 9, you
get the green curve and now you see that it
is looking more and more like the square wave
that you have met in the previous curve.
This is the square wave and you see that this
summation is looking more and more look it
is looking more and more like the square wave
of this picture. I strongly urge you to do
this, because all you have to do is to add
just a few terms and you can do this plotting
on a small graphics calculator if you like
or you can plot these graphs on a piece of
graph paper or you can do it on some computer
with a graphics plotting routine.
Now, here is another periodic wave which is
a triangular wave. This is a triangular wave
it is also called as a saw tooth wave, for
obvious reason it looks like a saw tooth.
This is a summation in which you have again
the odd multiples of the angle theta, so you
have sine theta in the first term, sine 3
theta in the second term, sine 5 theta is
next term and so on but, then the denominator
is not the corresponding n integer but, it
is the square.
So sine 3 theta divided by 3 square, sine
5 theta divided by 5 square, sine 7 theta
divided by 7 square and notice that the alternative
signs are different. This is the minus sign;
you have got a plus sign and then a minus
sign over here and a plus sign over here.
So, if you construct a superposition and for
different periodic functions the details will
be different but, I have illustrated this
for two functions to two periodic functions:
one is the square wave, the other is triangular
wave. You can see that you can generate any
periodic phenomenon in terms of a superposition
of just sine and cosine functions.
So, no matter what the nature of the function
is and if it is true for the square function
and for the triangular function for which
we could see the Fourier decomposition so
easily. It is in fact true for any periodic
phenomenon, no matter what the shape of the
pulse may be as long as it is a repetitive
phenomenon. In general means what we did was
to plot only a function of x but, a physical
phenomenon may be a function of both space
and time.
A pulse will be written as a function f of
x as well as time or more generally in three
dimensions as a function of the position vector
which will have three components x, y and
z or no matter what coordinate system you
are using. So, it will be a function of space
coordinates and the time coordinate and this
is typically called as a pulse or a wave function
or whatever.
So, it is and you can expect it to be decomposed
into sine and cosine functions, if it is got
a periodic element. Now, the periodicity can
be either with respect to x - with respect
to the space coordinate or it can be with
respect to the time coordinate or with respect
to both. In general, it is periodic with respect
to both, which is how the wave packets you
know traverse in space and time. So, you have
got in general a function of space and time
and you have these pulses or wave packets
which propagate with time and you can easily
do a Fourier analysis of these functions.
Here the basic solutions are coming from the
sinusoidal and the cosine function. These
were the solutions of our simple harmonic
oscillator, the solution was e to the i omega
0 t that was a basic solution.
What is that? That is the cosine term and
a sine term. So, the solutions of the simple
harmonic oscillator are fundamental to this
analysis, fundamental to the phenomenon of
oscillators, damped oscillators, damped and
devin oscillators, resonances, wave motion
as well.
Transmission of energy, electrodynamics, quantum
mechanical wave functions everything comes
under the application as a part of an application
of the basic analysis that we have learned
from solving the differential equations for
the simple harmonic oscillator.
Now, as I mentioned a typical wave packet
will have a superposition of different sinusoidal
waves; each sinusoidal wave will have its
own wave length, each sinusoidal wave will
be traversing at its own speed and as long
thus the speed of all these different components
is the same in a medium and that will depend
on the properties of the medium obviously,
right?
So if this speed remains the same, then the
shape of the wave packet will not change;
so it will traverse without any distortion
and this is the characteristic feature of
a medium which is non-dispersive whereas,
in a dispersive medium the shape will change.
You talk about the wave packets spreading
because it sort of it does not retain its
shape and it spills out of the original shape
that it is started out with.
Let us look at this wave function. This is
the function of z and t it is a function of
both space as well as time, so the space parameter
z or z and the time parameter is t . So, the
phase velocity of this particular wave is
given by this ratio omega over k or nu lambda
which is a product of the wavelength and the
frequency; the frequency is nothing but, the
inverse periodic time.
Notice that at a fixed z at a fixed position,
this is the harmonic function in time. So
as time changes, at a fix point the value
of this function will change harmonically
like a sinusoidal function or a cosine function.
Likewise at a fixed time, this will represent
a harmonic oscillation space. So, if you plot
it as a function of the space coordinate,
at a fixed time if you take it snap shot in
your camera.
At a particular instant of time, it will look
like this picture on this screen and this
is the obvious definition of a wave length
which is the distance between two points of
corresponding phases and then the peak here
is a crest and then the bottom here is the
trough, the maximum displacement is what you
call as the amplitude. So, these are important
parameters for this analysis which is the
frequency, the time period, the wavelength,
the amplitude and the phase.
Now, let us spend some time discussing this
phase. The phase function is omega t minus
k, this is the phase function. At a given
z, at a fixed value of z the phase varies
linearly with time; omega t is a linear function
with time, if you plot this angle as a function
of time it will be a straight line because
it is scaled by a factor omega which is a
constant and then the power of t is t to the
power 1.
So, this is a linear function of time it goes
as a first power of time and for a fixed times
it is a linear function of space because the
function of space so far this angle is concerned,
the phase angle is concerned is this kz. So,
for a fixed time this is the linear function
of z.
In the medium, if you look at how the surface
of constant phase would propagate then this
omega t minus kz, this phase will have to
be constant for all points on that surface
of constant phase. So this argument would
be 0, so d phi is equal to 0 and d phi is
the differential of omega t minus kz which
is omega dt minus kdz. So, essentially what
you have is omega dt minus kdz equal to 0.
So from this relation you immediately get
that dz by dt is equal to omega over k. This
is what gives you the phase velocity.
The phase velocity is given by through what
distance the surface will have to move in
a certain amount of time and that ratio delta
z by delta t, in the limit delta t going to
0 this ratio will be exactly equal to omega
over k and this is the definition of a phase
velocity for a particular wave. So, this is
the speed at which a wave front which is defined
by a surface at certain fixed phase. That
could be any fixed point, it does not have
to be the crest and it does not have to be
the trough, any particular phase that you
track but, you see that all how that particular
point propagates in space and time and this
will give you the corresponding phase velocity.
So, omega over k would be a constant and this
is the property of non-dispersive medium these
are called non dispersive waves. It makes,
it sounds as if this is the property of wave
but, it is more a property of the medium in
which the wave is propagating because whether
this omega over k will be a constant or not
depends on the medium rather than on the waves.
So, it is the property of the medium, so this
is like transferred a method in which you
have a description of the medium but, you
use it to describe the wave.
So in a dispersive medium the behavior is
not so simple because the omega versus k relationship
in a dispersive medium is not just a linear
straight line relationship. It is a little
more complicated, it depends on the wavelength
that you are talking about.
So, this omega which should be plotted actually
as a function of k and the reason it happens
is that depending on the properties of the
medium, you may have different dispersive
relationships. I will show you some of these
examples. When you deal with refraction; for
example, this is the kind of thing which leads
to refraction in a medium because the medium
disperses the different wave lengths, different
frequencies so that you get when light travels
through a prism for example, its spreads out
in different components.
So before we get to that, let us deal with
superposition of waves because typically a
wave packet will consist of several components
which are super post on each other. So here,
you have an example of a superposition of
two waves one for which the frequency is omega
1 and the wave number which is 2 pi by lambda
is k 1 and for the other the corresponding
parameters are omega 2 and k 2.
Let us assume for the sake of simplicity that
both have the same amplitude A. So, we construct
a superposition of such two waves and all
you do to work out the analysis is simple
trigonometric relations like the addition
and subtraction of cosine and sine angles.
So you can work out the algebra quite in a
simple manner and if you play with these terms
using ordinary trigonometry relations which
are the usual trigonometric identities, you
can construct the superposition of this by
writing. Thus cosine of these two terms and
then combining the corresponding terms and
you find that this net function psi the superposition
can be written as a single sinusoidal wave.
Now the cosine wave is also called as a sinusoidal
wave, the cosine function after all looks
just like the sinusoidal wave it is only phase
shifted by pi by 2.
So, it is still called as a sinusoidal wave
but, this is a cosine function. So you write
this wave function, this psi z of t which
is coming from a superposition of two cosine
functions as a single cosine function. That
is the net result of the superposition with
the difference that the frequencies and the
wave numbers are however different. The frequency
is neither omega 1 nor omega 2 and the wave
number k is neither k 1 nor k 2.
It comes as a result of the superposition
and these frequencies; so here, you have an
average frequency and here you have an average
wave number. So, the average frequency is
just the arithmetic average of these two frequencies,
average wave numbers is just the arithmetic
average of these two wave numbers. These are
the average frequency and the average wave
numbers that you get in the argument of the
cosine function .
Now when you construct the arithmetic average
you take the sum of the two and divide it
by 2 then, amplitude itself comes not from
the sum but, from the difference of the corresponding
terms. This amplitude is called as modulated
amplitude. It is neither A, it is nor this
factor but, it comes and it is modulated,
so this amplitude is A mod and this requires
another function which is coming from the
modulated frequencies and the modulated wave
number.
These modulated frequencies and modulated
wave numbers are coming from the differences
between the two frequencies omega. So omega
mod is omega 1 minus omega 2 divide by 2 and
the k mod is k 1 minus k 2 by 2.
So this is how now this comes from the plane
trigonometry, there is no magic over here.
We can now ask the question, you now have
a modulated function which is a sinusoidal
wave a cosine function and at what speed does
the modulated modulation propagate? Because,
this will be a function of space and time
and you can ask at what speed is this modulation
travelling? How will you find that out?
What you will have to do is to find, we look
at this argument v mod t minus k mod t z.
This is the argument and this must remain
constant. What is the condition for that is
what we will find and to do that you set the
differential of this argument to 0. So this
is differential of this argument z equal to
0 and you get a condition for the ratio omega
mod over k mod, so that will give you the
corresponding dz by dt.
So dz by dt which is given by the ratio of
omega mod to k mod which is nothing but, the
ratio of the difference in that two frequencies
as we have seen in the denominator we have
the difference in the corresponding wave numbers.
It is delta omega by delta k and this modulation
then travels at a different speed, this is
what you call as a group velocity. So there
is a difference between the group velocity
and the phase velocity. So, the phase velocity
is what we described earlier: this is the
phase, this is the velocity at which the individual
sinusoidal waves propagate and this is the
entire modulation, how it propagates at what
speed so that is given by the group velocity.
In non-dispersive media, the wave packet spreads,
so let us see an example over here. Here,
you have a case of refraction and you have
got light which travels along this ray and
then when it is crosses the surface of a medium
it bends, this is the phenomenon of refraction.
We ask the question, why does light have to
go from A to B to C? Why can it not go from
A to B prime to C? Why should it take this
the particular path, why should it not take
some other path? Why should it not come here
and then bend at this angle?
There is some reason why it takes a particular
path and this comes from the variation calculus.
This is the subject we dealt with the unit
1, in which we talked about the evolution
of the mechanical systems being described
not by the principle of causality. But by
the principle of variation, namely the Hamilton's
principle of variation, we argued that a system
evolves in times such that action is an extremum
and in this context it translates to what
is called as the Fermat's principle. The time
taken by light to go from A to B to C rather
than A to B prime to C for any other path
through some other point B prime anywhere
on the surface, compare to any other path,
this will be the least time.
So, this is the principle of least action
and you can see it very easily because all
you have to do is to write the expression
for the time taken for light to go from A
to B and from B to C and the sum total of
this is the total time taken by light to reach
C through the point B. What is this? This
time is divided by the distance by the speed
of light and the speed of light is not same
in the two medium.
So, it is v 1 in the medium 1 which is the
upper medium and it is v 2 in the lower medium.
So this time is the distance over here, which
is given by the sum of the squares of this
distance and this by the Pythagoras theorem.
So this distance is the square root of a square
plus x square divided by the time by the speed
in that medium which is v 1 and then over
here the speed is different, which is v 2.
This is the hypotenuse for this right angled
triangle and you can see that the corresponding
distance is from this green line to this green
line is d and this to here is x. Then, the
distance between these two green lines the
second and the third green line is d minus
x .
So, you have to take the square of this distance
and this distance is b, so b square plus d
minus x whole square and then you take the
square root to get the length of the hypotenuse
itself divided by the speed of the light in
medium 2 and this gives you the total time
traverse that is required for light ray to
go from A to B to C.
You can ask, what will be the change in this
time if the ray were to go not through the
point B but, through some other point like
B prime. If it were to go through some other
point B prime what would change it is this
distance x because when it goes through the
point B; this is the value of x which I am
pointing out by this pointer whereas, if it
goes through the point B prime this will be
the corresponding distance. So, this distance
is a measure of which point of the surface
the light ray will go through, so if you take
the derivative of the time taken by t with
respect to this measure which is x dt by dx
all you do is to take the differential of
the right hand side with respect to x.
So that is simple algebra, what you do is
take dt by dx, find this out, solve it, no
need to write down these terms in your notebooks.
All you have to do is to take the derivative
of t with respect to x and then demand that
this derivative is 0, because by Fermat's
principle, it is going to take the least time,
so at that least time this derivative must
vanish.
So put this equal to 0 which means that these
two terms, this minus this and this difference
must vanish. You will get a relation for the
ratio of v 1 over v 2 and that is exactly
what you get because this ratio will turn
out to be given by the trigonometric properties
of the right angle triangle that you have
constructed. You get a very simple relationship
that the sine theta 1 upon sine theta 2 is
equal to the ratio of the two speeds and this
ratio of the two speeds is what the refractive
index of the medium is.
Now that we know what the refractive index
of the medium is, we understand dispersion
very easily by recognizing that this refractive
index actually changes with frequency is not
the same for all the components in white light.
Because this refractive index changes for
different frequencies, then you have the different
waves going at different speeds in different
media. Since, they go at different speeds
then, they are going to spread out.
So that is exactly what happens and this is
what generates for us what a dispersive medium
is and you find that at the bottom of all
of this is the simple algebra of simple harmonic
oscillator. It is for this reason this is
one of the 100s and 1000s of examples in physics
and engineering why we learn this. It is not
our intention here to study the dynamics of
wave packets or the Fourier analysis or optics,
let alone quantum optics or any details but,
just to give a flavor of the applications
of the range of phenomena for which a very
clear and comprehensive rigorous understanding
of the oscillator and the damped oscillations
and the resonances and so on is important.
Now, this is the phenomenon of dispersion;
you find that this refractive index changes
with the frequency, so different wavelength
will disperse when passing through a medium.
This is the famous experiment by Newton and
then you have the red at the top and blue
at the bottom this is how the wave is spread
out and this is what is called as normal dispersion.
There are peculiar properties of different
media and you find that the refractive index
is given by the ratio of two speeds, so it
is a speed of light and vacuum, it is the
speed of light and the medium; the speed of
light and vacuum is usually written as the
letter C, which is a universal constant.
The speed of a wave as we have seen earlier,
when we discuss the phase velocity is nothing
but, the product of the frequency and the
wave length. So, the product of the frequency
and the wavelength is this nu lambda but,
the lambda will have to be different for the
same frequency. Because these two spaces are
different, the refractive index is not equal
to 1, only when the refractive index is equal
to 1, these two spaces will be equal and the
numerator and the denominator will be equal;
otherwise the two spaces are different and
the two wavelengths will be different. The
corresponding wave numbers are different because
the wave number is just the reciprocal wavelength
2 pi over lambda which is what defines the
wave number.
So this is the ratio of the wave number for
the medium divide by the wave number of the
vacuum. So in a medium, you will have omega
which is given by the v medium divided by
lambda times 2 pi of course, because you are
writing for the circular frequency.
Now, you know that this v medium over lambda
medium if you just swap this term from this
relation over here, this will come from dividing
the speed of light by the refractive index.
Because the refractive index change depends
on frequency the medium becomes dispersive.
So it is the medium which is dispersive but,
you often say that the waves are dispersive.
So now you can see if you plot a graph between
omega and k omega versus k, you plot k on
the x axis and omega on the y axis then, omega
versus k is related by this particular factor
proportionality. Now if this proportionality
were same for all the frequencies you will
of course, get a linear relation but, if this
proportionality changes with frequency you
cannot get a linear relation. Now this is
the characteristic feature of a dispersive
medium; the omega versus k graph will not
be linear because for different frequencies
you will have a different proportionality
between omega and k.
Now, this is exploited in modern physics and
I am not going to be able to discuss this
at all but, physicists play with this and
they are able to control the speed of light
actually bring it to a halt depending on how
you control the properties of light propagation;
this leads to very fascinating applications
like electromagnetic induced transparency
and so on. These are all specialized topics
in quantum optics and this is only to excite
some of those who will be interested in some
applications.
But the properties are very fascinating, you
can also find situations in which the here
is an example of a paper which was published
in nature about a decade ago. This was an
experiment carried out at Princeton by Wang
and his teammates in which they manage to
get a laser pulse travels more than a 300
times a speed of light. Now, it does not mean
that you are violating any fundamental rules
in physics.
How all this reconciles with the fundamental
rules in physics is a matter of detail, it
is very fascinating; you have to deal with
very complex dispersive phenomena; there is
this normal dispersion, there is a anomalous
dispersion, in which the refractive index
actually decreases as the frequency increases
not increases but, it decreases; that is the
case of anomalous dispersion. So what you
will find in an anomalous dispersion is that
if light were to go through a medium which
has got anomalous dispersive properties, the
ordering of the red and blue will be reversed.
There are even more fascinating materials
which are like meta materials in which you
even have negative refractive index and so
on, so that is something for you to read about.
I will leave you, with one picture before
we conclude this unit. This picture taken
from the maid of the mist boat ride near the
Niagara Falls and this is the picture of a
lovely rainbow. This is taken because there
is a mist and the fall of the Niagara Falls
and here I want to leave you with one or two
questions for you to ponder over.
Questions which probably come to your mind
are ready on seeing this picture. Why does
the rainbow have red outside and blue inside?
The other question is which part of the picture
is the brightest and why? So this is not going
to be a part of our discussion today but,
these are the questions I would like to leave
you with, if there are any questions.
Rainbows is always such a lovely sight; in
fact sometimes you see a double rainbow and
then you have to ask yourself what is the
ordering of red and blue in the second rainbow.
But I let you worry about these questions
and the answers come from the simple phenomenology
that we have talked about together with some
simple techniques that you use in and doing
optics; it comes from this essential consideration
that the refractive index is frequency dependent.
I will give you as a hint that this refraction
of course, is in water droplets. So water
has got a refractive index which is different
for the red and different for the blue; for
the red the refractive index is about 1.331
and for the blue it is 1.343. So they are
not exact, they are slightly different and
the answer lies in this, the details are for
you to work out.
I guess, I will stop here and then next time
we meet we will go over to unit 3 in which
we will learn about various coordinate systems,
we will begin with the plane polar coordinates
and then the cylindrical polar and the spherical
polar coordinate systems. We will learn about
the dynamical symmetry of the Kepler problem
and various other things as the course will
progress. So, thank you all and we will conclude
unit 2 and begin with unit 3 the next time.
