Optics (Lec - 12) - Electronics & Communication Engineering Video Lecture - Electronics and Communication Engineering (ECE)

the following content is provided under
a Creative Commons license your support
will help MIT OpenCourseWare continue to
offer high quality educational resources
for free
to make a donation or to view additional
materials from hundreds of MIT courses
visit MIT opencourseware at ocw.mit.edu
okay so let's get started I understand
to wire myself have you okay so hmm so
before we get into the into the to the
PowerPoint here so basically what I'm
doing is is I'm picking up from the
point where we left last time we were
discussing waves and what I would like
to do now is derive wave equation
basically for a mathematical form that
there may that allows that should allow
us to predict different kinds of waves
of waves not the way all right so what
what we start with here I felt with a
general function C of an argument at so
for those of you who are not not fluent
in the Greek this is C this is hat okay
and and these are at the moment this
don't necessarily have the physical
meaning but in order to produce a wave
out of this function what I can do is I
can write it as C or F of Z comma T
equals C of Z minus CT okay now why is
that a wave if you first of all Z now
has become a special coordinate T usual
is time so if you actually hide time so
now time is 0 or time is a constant or
something like that then you actually
have basically the same argument but if
you allow time to increase then you can
see that it the time increase will have
the effect of sliding the
the same say along the special axis so
for example if let's say let's take this
as our reference point here the first
peak after you advanced I'm a little
further then this point has moved fast
has moved to the right and how much is
this distance well anybody this is easy
I just I'm just trying to see if the
people in Boston working up
yes
nobody's awakened boss so this distance
equals city right because if I if I set
the argument of of this function to zero
then basically recover my original
function so so basically as a advance
time the effect is that this waveform
moves to the right okay so now what we
do is and at the moment there is no
obvious motivation for why I'm going to
do what I'm about to do but please take
my word for it for a while I'm going to
take two partial derivatives I'm going
to take DF DZ and they have the team
okay the FD z equals basically C prime
were C Prime I used C prime is simply to
denote the derivative of C with respect
to its mysterious argument so that's
very easy what about the F DT
that's right I have to apply the chain
rule because there is a multiplicative
argument inside so this is going to be
minus C times C Prime
so because C prime appears in both in
both places over here I can combine
these equations and say the F of D Z
equals minus 1 over C be our bitting
okay what is a partial differential
equation and you might say well what I
did is a little bit stupid right because
I had a relatively simple expression for
the wave I actually had the wave just in
front of me and what I did was I wrote a
partial differential equation could but
the benefit of doing that is of course
that the wave that they wrote here is a
very special very limited kind of wave
that propagates without change in shape
without anything fancy happening we know
from experience for example in real life
if you have ever seen water waves water
waves they don't quite propagate like
this they change shape in all kinds of
complicated ways so even even though I
derived this equation from that simple
set we will see in a little bit later
that this equation actually can describe
much more general wave we're not quite
done yet because the question that I
derived here applies for a wave moving
to the right
I can equally well consider a wave that
is moving to the left and in this case
what I would have to do is I would have
to write F of Z comma T equals C of Z
plus city a little bit of thought will
convention out that if you advance time
then the way we will have actually to
move to the left towards negative Z in
order to maintain its argument and if I
do the same game now we can simply again
I can write DF be Z equals C prime that
did not change this time the F DT was
Plus C C Prime
now have another partial differential
partial differential equation we have DZ
equals 1 over c DF DT ok let me put back
the question so this is a wave going to
the left backwards and had another
equation VFD Z equals minus 1 over C the
DT that is a way of going to the right
ok you can see very easily that if you
take a second derivative of these
equations you would actually just go
back to the first one if you do the
second derivative d square F this is
square we will pick up an extra minus
sign from the same equation that you
will find that it is 1 over C Square D
square F DT square you can do the same
in this equation there's no sign here to
begin with that is 1 over C Square D
square F DT square so basically this
single equation
the single equation can describe both
forward and backward and propagating
waves and this equation actually it's a
three-dimensional version of this one
but its three-dimensional version it is
it has the name Helmholtz wave equation
and because of Helmholtz it's a little
bit difficult to pronounce actually
that's not the reason but the legend is
anyway it's a very commonly encountered
wave equation
so typically we simply omit Helmholtz
and we say wave equation now why we
might want to distinguish this as
opposed to are there other wave equation
yes there is the Schrodinger equation
that is the klein-gordon equation that
is the for those of you who do fluid
mechanics there is a quarter deck the
bridge equation also known as kdv
there's a bunch of different wave
equations so we need to we need some way
to discriminate nevertheless the other
wave equations they are referred with
their name so when you wanna refer to so
I think it is a question you say
Salonika this equation when by
convention when you say wave equation
you mean this one so the Helmholtz okay
but it's all very nice but and it's
actually done on your slides so there's
no reason to to to organize over what I
did over here what I would like to
remind you before I move on is for the
simple case of a first of all but you
can imagine that this is a very general
kind of solution because what is see
after all I put up this this strange
shape over here I didn't say where it
came from so for that matter if someone
gives you the wave equation how can you
go ahead and solve it well we're not
going to do this in great detail here
this is a little bit beside the scope of
the
but generally if you need to solve the
partial differential like this one you
need additional information you need the
initial conditions how did the wave
start at time equal to zero and you also
need the boundary condition so what
happens to the wave has it expands
sometimes the boundary condition is at
infinity maybe the wave is not bounded
it can propagate outwards all the way
and that's a good example of what
happens if you are you know flying with
a helicopter on top of the ocean and you
drop I don't know you drop a stone into
the surface of the ocean then the wave
that you generate the wave you that you
generate to a good approximation you can
use infinite boundary conditions right
because the ocean is not quite infinite
but very large if if you generate in a
wave in a small channel then you might
have to include boundary conditions
right because the wave will be confined
by the walls of the channel and we will
see later that this is actually quite
commonly the case in optics we don't
call them channels we call them wave
guides but but the boundary conditions
can actually significantly alter the
shape of the wave so what I mean here is
just a resort to give you an idea a what
a called solved nothing interesting
happens here but this is actually a
solution of the wave equation with a
particular initial condition where at
time equals 0 the wave is a sinusoid so
so I basically said I said that this is
my equation and I say that F of Z at T
equals 0 equals some sinusoid with
amplitude a and a wave number K
if that is the case then all by
inspection I don't have to do anything
very important and what I did not write
down but it is implied is that the
boundary conditions are free so this
wave is free to go as far as it pleases
without any special constraints so if
that is the case then then you can write
the solution almost by inspection f of Z
comma T equals a cosine KZ by K Z minus
CT and you can see by inspection here
that because of the way I wrote it this
actually satisfies both of the partial
differential equation the wave equation
and the boundary condition so therefore
this is a proper solution of the of the
wave equation in fact I only wrote half
of it the wave remember can go forwards
it can go backward so to cover both
cases I actually write the solution this
way with a data that I have here
actually I don't have enough to specify
actually in principle the wave can go
both ways I would have to apply to give
you additional information to know which
actually happened the other thing that I
wanted to point out which is not really
difficult but if X I'm getting used to
is the various terms that appear inside
this equation over here so the terms
that we've been very familiar with is
the wavelength there's another over
those crazy Greek symbols that called
lambda and the wavelength is related to
the quantity that we have over here as 2
PI over lambda equals K so lambda is the
wavelength as I said K is known as the
wave number so so the relationship that
they have that they have is if you think
of the wavelength as a period which it
is the period of the wave in the space
domain then K would actually be
the angular frequency but we call it
wave number to keep them separated okay
equivalently you can define the actual
angular frequency of the wave usually
this goes by the symbol Omega and from
this equation over here very simply you
can see that if you wanted to write it
in the form a cosine KZ minus Omega T
then clearly Omega is the angular
frequency and be comparing the two
equations you can see that Omega equals
KC so the angular frequency of n is
related to the spatial frequency the
wave number and the and the frequency
and the and the I'm sorry and the speed
over the wave and of course this is
nothing new because we've seen this
equation before but we saw it in this
guide and that's why I'm going through
this because all the symbols they can be
played around with quite a bit so this
equation I can rewrite and the very
light is like this so Omega is the
angular frequency we also have the plane
so to speak frequency nu which is
related as Omega over 2 pi okay
one of them goes full the radiance
I mean I'm sorry is better than in full
radians per second this is Omega nu is
measured in health simply inverse
seconds okay pleasure
so so I can substitute in this equation
Omega is 2 pi times nu K according to my
question over here is 2 pi upon lambda
and C is C so the two PI's cancel here
the to pass cancel here and I end up
with the question C equals lambda nu
which I think of what are they seen a
couple of times before who call it the
dispersion relation of the wave okay so
this is how all this come about
I haven't said anything new
about this I'm just trying to point out
the different ways that this equation
can Menace asthma can manifest itself
okay and the other two terms that we
encountered last time and I wanted to
remind you one is the amplitude of the
wave this is this constant a over here
for some reason in the slide that I
decided to call it a knot anyway it is
the same thing and also we said last
time that in general the wave can also
have a phase delay so in here i can also
add the constant plus fee which does not
really do anything by itself all it does
if you wish is equivalently it shifts
the time origin so that's a relatively
trivial thing because after all i am
free to start my clock whenever I want
if I have a wave that never changes I
can start my clock whenever one so it
doesn't really make a big difference but
we saw last time that if you have two
waves that are as we said interfering so
there are simultaneously happening so to
speak then relatively the relative phase
delay between the two can actually be
quite significant and I will play this
movie again in this case the two waves
are actually they have the same phase
between them in other words they if you
look at the center they're both
simultaneously bright for simultaneously
dark whereas in the other case it's the
other way round they're off by PI so
when one is maximum the other is minimum
I think it is worth playing this one
this also has the interesting fact that
it can happen in hypnotized you so for
those of you who are in Boston are not
fully awake yet yes oh so when you are
requesting you to push the button
so far dampen our two weeks travel in
the opposite direction they have a sim
amplitude but perfect cancel each other
so in this case where does the energy go
not quite cancel each other right
so as we will see I will actually do
this in a second it's called the
standing wave so you will see that they
don't quite cancel they they just
produce it well what is called the
standing wave so I will get to your
question and maybe I think is actually
coming up now yeah so give me a second I
will get to your question okay
so that is a standing wave so like your
colleague asked over here suppose that
you have a superposition of two waves
now both are solutions to the wave
equation now by the by its looks I don't
want to do the proof here it is it takes
three lines but by its looks this is a
linear differential equation so
basically it means that if you know one
solution you know another solution and
then you add these two solutions after
multiplying by arbitrary constants the
result from the addition is still a
solution to this equation very
fundamental property of that we use
again and again later in the class so
that said suppose that I pick two
solutions that they already saw one of
them is a forward propagating wave the
other is a backward propagating wave so
here they are
the one with the minus is propagating
form work the one with the plus is
propagating backward so your colleague
here asked what will happen if I
superimpose them so one of them is going
to the right one is to the left are they
gonna cancel what are they gonna do okay
so this is a computational so perhaps
you should not trust it after all
what is this but anyway this is what I
got in MATLAB when I simulated this okay
let me play this again so this is the
result of the two waves
okay I pushed the button I did not let a
movie again I revealed the derivation
but that's quite okay okay so I will
actually meant to do this on the one the
free hand here
okay nevermind ignore what you see on
the on there slide then just watch the
the whiteboard mmm I guess I can call it
white board can't I and it is a
whiteboard
okay okay so so here the two waves
so I actually omitted the should have
said that not only are the
counter-propagating but also I pick them
both to be harmonic waves so that well
so that would who can do the math easily
okay so let's write this out so we
picked the exact the identical amplitude
so one of them will be written as did
you say Onizuka failure okay
okay now for sim for generality I don't
really have to but for generality I also
added the face face delay fee to the
first wave so this one is forward and
then I have the other one which looks
very similar except for this devious
minus sign that goes backward okay and
the superposition principle says that
they can simply add the two waves and
the summation is still a solution to the
wave equation therefore still a valid
wave okay so what is the result so now I
don't know how how to tell you to the
memories
most people look it up in books of
trigonometry I have a certain mnemonic
for it but I will not tell you but
anyway so there's a remedy monitor says
if you have the sum of two cosines
equals now this is a mathematical
property so there's no intuition to be
derived here this is this is obtained by
trigonometric manipulation equals twice
cosine a plus B over 2 cosine a minus B
over 2 okay so I never remember this one
I will tell you actually I am my
mnemonic is not is not that in strange
all I remember is that if you have
cosine of a plus of two things
it equals let me write it in your own
linkage so cosine of two things
it has a minus sign here and sign or two
things
okay these two I have memorized and you
can see very easily that from this two
you can obtain the question we said
before because for example you can write
cosine alpha plus that plus minus beta
equals cosine alpha cosine beta minus
plus sine alpha sine beta okay so if you
add the term with a plus and the time
with a minus the sine terms of a polite
and you end up with with a identity that
they're before so this is Magne monic
for remember this one we will not have
to do this too often because the only
last time if you remember who introduced
phasers going to reduce the complex
notation for waves so now we don't have
to deal with this trigonometry very
often in this case there's no way out
because well actually there's a way out
but it is simpler to do it okay right so
now I have basically a equals KZ minus
Omega T plus V B equals KZ plus Omega T
plus V so if you apply my formula here
you will find that the wave actually
okay the amplitude is the same I pick up
a factor of 2 so it is 2 and then I have
the cosine of the sum so if I sum the
two the time param will disappear so
we'll have cosine and that is well you
get twice of it but there's a factor of
2 years of the console will be cosine
cosine KZ plus V and if I subtract the
two then actually the special terms and
the phase will disappear and I will get
a cosine Omega T okay now do I believe
MATLAB I think I should believe MATLAB
because let's see what did MATLAB saw
here time is frozen so you see basically
the same term where this has been
replaced by constant because the first
time so indeed this looks like a
sinusoid of the same frequency as we had
before
and actually of course you have no way
of knowing that but the waves that they
entered had an amplitude that was half
of this so basically the factor of two
was also reproduced by my simulation
okay now if I play the movie again it is
basically like and freezing time okay so
here I have unfrozen time and you see
that I get a sinusoidal variation in the
time domain so therefore that's my term
cosine Omega T here so basically the
MATLAB simulation is correct we were
very happy okay so now then let me come
to your question we have what happens to
the wave when the when it vanishes right
first of all it doesn't really value Sri
I don't know how to do
okay
so that look like at some point the wave
vanishes and then it reappears okay
there's a little bit annoying that it
vanishes but in electromagnetic waves
and usually in other kinds of waves we
won't worry about the instantaneous
energy we'll worry about the average
energy so what you can see here is that
the average energy is actually concerned
because as the wave oscillates from a
maximum to a minimum its average energy
remains the same
now you might ask well what happens in
between I mean you know what happens
when the when this wave has gone to zero
again a will a central cause of
electromagnetic waves they well the what
we have done here is we have basically
launched two waves that are moving in
opposite direction today so who is
providing the energy well if you are
providing the energy on one end and
someone else is providing the energy on
the other end so what happens when what
happens when the two waves cancel is
that the energy is actually stored in
another place which does not show up in
this equation that's one way to to think
about if you if you remember when I saw
actually you don't remember because you
were not in the class one week ago but a
certain example of a simple harmonic
oscillator right so simple harmonic
oscillator goes from zero velocity
maximum potential energy to the edge
where the velocity now is employed to
the left position where the velocity is
maximum and the potential energy has
become zero right at the last position
so the same kind of thing the kinetic
energy disappears well what happens to
it it has become potential but is very
similar to this the energy has become
potential energy and you don't see it
here yeah
okay maybe your follow up on the
question so now let's assume the two we
were traveling along the same direction
and that they are out of phase by PI so
music in this case they be a perfect
cuts on each other other so where does
energy go just for your information the
previous answer I get is such Simula not
happen in the first place but just not
comfortable with this answer so I would
like to know your comments well either
the wave started by phase shift it's
equivalent to that not having launched
anyway the Tollan whoever launched the
two waves basically cancelled his own
action so if you think about it
physically what happened ray first of
all had the two waves been launched at
infinity or at the finite time so for
example now and lunch and wave and a Z
equals zero and this is a way of
traveling through the day as to the
positive direction and continuously now
I'm not sure another wave and as M
equals of a number years that equals
minus ten but this show it is we've only
last for three period for example so as
this way this way of travelling into a
positive direction it's going to cancel
three periods our previous wheel right
now a bit more complicated because what
happens to the edges of the wave are you
multiplying by by a boxcar function what
happens later right
it's not you're not given enough
information to answer the question right
you just tell me that three periods
cancelled but then what happens later
did you just stop the wave or it's not
understand or sucia yeah that's right
I'll check yeah thanks the questions are
defined again so I cannot answer it but
but I can tell you this suppose that I
launched a wave at some time and then a
certain time later I launched the same
way with PI phase shift rate so there's
no doubt that they would - waves at this
point will cancel so what happened to
the - the energy well it is equivalent
to thinking that they did not really
lose launched a second wave with a PI
phase shift but they actually turned off
the sweeps that was producing my first
wave in the first place okay so with
there might be a transient or something
like that but in effect I've just
canceled the wave so simply stop the
flow of energy right I mean that that's
not something that would be happy with
this answer or discuss this
okay
nothing today Eliza actually this is
interesting there they brought up energy
in the standing waves the energy is
actually not going anywhere right
because depends well that's one way to
think about it it's not going anywhere
or you can think about it in the way
that the Energy's going back and forth
right so it's yeah okay
well actually I did not know Fraser in
last time you do that okay
right so so yes I pulled the first one a
few minutes ago I was talking about the
difficulty of trigonometry and I said
that's why I introduced phaser last time
I was mistaken I had not introduced any
phasers last time I'm introducing
phasers now and illusion of course that
were introduced in phases because we
don't want to have to remember
complicated identities like this one
okay anyway another point that I would
like to bring up before I move on to
phasers is that the way that I'm showing
here is actually very different than the
travelling wave that led us to the wave
equation you agree with that that looks
very different both in the way it
behaves with time and space and the way
it looks like an equation yet it still
originated from the same wave equation
so this is actually the power of partial
differential equations and superposition
that I took two solutions of my partial
differential equation that were not
particularly interesting I superimpose
them and then a new value and then I
created another solution which is still
a very valid wave but we have totally
different it is a stationary wave it has
this oscillation and so on and so forth
so that is why we bother to write this
partial differential equation because it
allows us to generate a much richer set
of waves that of course very importantly
practice standing waves happen all the
time
okay so now let's go to phasers okay so
just like before I wrote two waves that
were propagating in opposite direction
now what I'm going to do is I'm going to
write two waves that are phase shifted
by PI over two one of them is cosine KZ
minus Omega T plus V okay
and the other is phase shifted by PI
over 2 which means instead of the cosine
that is a sign
okay both are very valid solutions to
the wave equation with different initial
conditions but that doesn't bother us
they're still valid solution therefore
the superposition is also a solution and
I said before that when you do a linear
superposition you can also multiply the
waves by arbitrary constants I said this
little bit fast but I hope that remember
that I said in fact in this class we
have evidence of what I said and what is
MA say and go back to the video and see
if I said it and I did okay
so I will pick a very particular
constant which may sound like a crazy
choice but I would pick it and that's
the imaginary constant I've also known
as one of the two square roots of minus
one okay what is this now so the first
confession that I will make is that it
is not physical anymore there's no such
thing as a complex wave the wave that we
see around us are real they can be
measured by instruments such as
oscilloscopes cameras know what else the
real the as far as I know nobody has
ever measured anything complex however
what I'm doing here is mathematically
correct even if it has no physical
meaning and it turns out it provides a
huge mathematical convenience so I'm
justified in doing it as long as I
remember what is the physics that is
behind and the physics that is behind it
is that every time I see a wave like
this I have to remember that the
physical quantity is the real part okay
so the physical wave the one that they
can actually measure observe generate
with you know natural means is the real
part the imaginary part is something
that I added for mathematical
convenience and originality is because I
forget whose formula is called the MOI
world
anyway one of those friends de moivre
yeah the Moapa yeah okay I go to travel
so the mother then if I remember
correctly the scariest family the mother
says that this thing here is a complex
exponential okay no that's very nice
because for example if you have to
multiply two cosines it's a big pain
that you have to remember the trick to
multiply two Exponential's it is trivial
you just add their faces I mean they're
their exponent so so a lot of a lot of
convenience is to be to be expected so
this representation it comes with
different names which sometimes used
confusingly in an interchangeable way so
some people call this a phaser notation
I prefer to call this the complex
representation of the wave it becomes a
phaser when we drop I'm going to do
something nasty here okay remember that
I can i this is a product right so
Canada says e to the i kz e to the minus
I Omega T e to the I Ching okay if you
recall during the beginnings of the
class we saw that the wave length in a
medium changes but there is a property
of the wave that does not change that
property if you recall is the temporal
frequency Omega or new it's related by 2
pi okay that is true in linear media of
course there is a class of optics called
nonlinear where actually the frequency
in changes can double or even
or fancy things can happen to it but in
this class we don't deal with this
complicated things we'll just deal with
linear optics so in linear optics the
temporal frequency of the light no
matter what happens to the light even if
it goes through glass and reflects and
all kinds of things that important
frequency remains the same so when we
deal with optical waves of a single
frequency that is the corner is fixed
and the medium is linear then this term
is actually super full we don't have to
remind ourselves that the light goes
like e to the minus Omega T it does but
we don't have to write it because if we
keep writing it it clattered is our
equation so we simply drop it okay drop
it very important to remember we allowed
to drop it if we know that we have a
single color in our system a single
frequency if there is multiple
frequencies we cannot do that anymore
right because then this these
frequencies have to adapt in a certain
way and we have to keep track of their
relative phases so okay so we can only
drop this term if we two conditions if
you know that we have a linear medium
and a single frequency light propagating
in that medium okay so in my terminology
at least after we drop this term then it
becomes a phasor so what is left e to
the ikj plus five that I call a phasor
and you notice what happened there is I
only left the spatial dependency of the
wave plus the phase delay the temporal
variation is it's hidden it is not gone
it is just hidden okay okay now let's
see this in action so what I'm going to
do is I'm going to derive the standing
wave but using the phasor so you can see
how nice and simple it comes out and we
don't have to agonize over it but before
I do it I actually have to solve a
little problem the problem is that I did
this very nice derivation for the phasor
of the forward propagating wave what
about the phasor for the backward
propagating wave okay
the backward propagating wave would be
something of this sort so this will
always forward a canal would let me
write the backward I think I have enough
space here so I can do it okay
so the backward of this would be
something of the sort of the form cosine
KZ plus Omega T last fee then it is very
inconvenient because the way I got the
failure is a got rid of the term of the
form e to the minus Omega T if I add the
imaginary part here I don't have a term
e to the minus Omega T some stock but I
can do some mathematical trickery now
this is sure tickly but it works the
trick is the following we know that the
cosine is an even function so this is
actually identical I can replace it with
in other words with the cosine of the
negative element not harm done it is
ticular backward propagating wave
nothing has changed is the same equation
now according to my good principle here
I
that is not the same as before right
because the sine is an odd function but
this is a non-physical term anyway I
just put it there for mathematical
convenience so don't particularly care
I'm very happy because the real part is
the same as I started with what
imaginary part does as long as
mathematically convenient I couldn't
care less right okay so that gives me
the license to write this now this of
course equals e to the minus I K Z minus
I Omega t minus V now very happy because
I did get a term it was a minus I Omega
T in the phase of this exponential and
therefore I can drop it to produce my
fashion so the lesser then is that the
phase or for the backward wave is e to
the minus I K is e minus IV in other
words if I give you a forward propagated
wave and you compute its phasor and then
I ask you well what is the phasor for
the backward propagating wave well all
you do is you flip the sign on the
exponent okay fine so in order to have
this can will produce whoa what did this
come to I think this is an old all the
paper from some previous lecture but
anyway then we ignore this I have to
remember class for in you five for less
than okay okay so what I will do now let
me reproduce the standing wave so the
standing wave is using the phasers now
is e to the ikj plus fin that is the
forward wave plus e to the minus I K is
e - I think that is the backward wave
can you deliver that well yes actually
because again the more ever says that if
you add two Exponential's of this form
notice they have the same phase within a
minus sign I will simply get two eyes
cosine is Z plus V so I got my my thing
here not quite though because the
standing wave it had the cosine Omega T
term what happened to this cosine Omega
T what happened to the cosine Omega T
well remember we neglected and e to the
minus I Omega T you didn't really
neglect it we hit it because we didn't
want to carry it when you arrived the
question well if I did if first of all
if I don't care about that important
variation I don't really have to do any
anything I'm done but if I want to put
that important variation back in what I
have to do is I have to put back what I
took
when did not belong to me and what did
not belong to me is did this e to the
minus I Omega T okay I put it back now
that is still not a physical wave
because it is complex how can I find the
actual physical way when I have to take
its real part the real part is easy here
because the only complex thing is this
exponential so if I take the real part
of this of all of this now it would
simply be 2 cosine KZ plus V cosine
Omega T and I'm done I've actually
reproduced my standing way
any questions
we're almost done here so I have about
three minutes or so I think I will use
them to finish up this lecture so can
move on bigger and better things
the question that I wrote before is the
one-dimensional wave equation because
the waves that I've written they can
only go down one special axis right Z
the more generalized
of course situation is three dimensional
waves and what you see here is the
generalized three-dimensional wave
equation which of course I will not
derive but it can be pretty easily
derived we using the same arguments that
would it before except you have to kind
of generalize them if you dimension and
this produces things that we've seen
before in geometrical optics for example
the plane wave except now you see the
wave description it is you have
alternating was positive and negative
how do you call those peaks and traps
which are propagating along an arbitrary
Ally now they can go in any angle in 3d
space right this particular so you see
just a cross-section here or you can
have a spherical wave which you saw it
before it starts at a given point and
then radiates outward that's a spherical
wave well we're not quite finished yet
but I think we're almost out of time so
I think I'll stop here we can still
continue next time are there any
questions
there's no homework due on Wednesday so
what I would like to do you have a
question oh yeah there's a question
right how is the wave equation
intuitively different from simple
harmonic oscillator like the variation
in space depends on the variation on
time
so that's coupling okay yeah the
mathematical answer of course is varies
if they simply have more nicosia later
is an ordinary differential equation
this is a partial differential equation
but let me let me think how to formulate
the physical answer okay so the physical
answer is that the the harmonic
oscillator is a single particle whose
position you track as a function of time
right a wave is a distributed physical
system so you can think of a way in this
case for example you can interpret the
physical meaning of these black and
white stripes as particles which are
moving for example the perfectly that is
on the white stripe is moving up the
particle in the black stripe is moving
down right whereas if I play the wave
again what you will see that the part it
is I can home because a Tooting like
coordinated motion right because you
have particles that were down then they
go up then they go down again but they
go out in a coordinated motion
throughout the entire plane how do they
know well there is a physical system
associated with it for example this
could be a membrane with the sound wave
propagating on e3 and now the reason the
particles are connected that the
executes the coordinated motion is
because of the entire atomic you know
forces between the membrane particle if
it is an electromagnetic wave then it is
a field that is that is distributed and
of course it knows to be connected
because of Maxwell's equation right you
know that things like charge
conservation and so on that force it to
act this way so a wave is actually
distributed system where the physics
force an extended
systems particles or fields or you know
whatever the case may be to execute the
coordinated motion that's the difference
between they simply have no notion it
over you just have one particle now you
can go from one to the other imagine you
take a single harmonic oscillator that's
a good one so let me let me draw it here
well we're actually going backwards now
in in the knowledge that they use in the
path so you take a simple harmonic
oscillator here it is right at the
pendulum right this is a simple harmonic
oscillator then you couple it with the
second pendulum how do you cut they're
not traveled at the moment but you can
couple them for example by connecting
the spring in between right but there's
not quite a wave but you can see now
that because I coupled them if I kick
one of these it will also cause the
other one to move right so they program
coordinated well now let me generalize
imagine that they have a bunch of
harmonic oscillators and in fact I can
make an infinite and a couple them now
in fact the system will become wave-like
in the sense that if you kick it at one
end okay let's say not at infinity let's
say it starts actually here and then it
goes to infinity so if you kick it at
one end then the disturbance will
actually propagate very similar to a
wave it's a discrete wave but silly wave
so you can go from one to the other by
introducing the kind of a coordination
mechanisms which is the coupling here
does the coordination
another question we can stay here
actually nobody claimed the classroom
say they here there so we have more
questions we'll be happy to stay in
answer them
but about the Pfizer's is it clear the
phasers I realized the phasers when I
was your age you went over the student I
was also very upset by phasers because
they they look very much like
mathematical tickity I mean I'm trying
to find my I guess that's one benefit of
this over the whiteboard that they have
a history of what they do but yet but
the phasers they look very much like a
mathematical pre-k I mean just look like
someone pulled it out of the Hat but but
the only reason to justify it in your
mind is that it provides huge
mathematical convenience now this looks
like a trivial thing I mean the
derivation of the standing wave was
maybe four lines with trigonometry two
lines with the complex exponential so
okay why mess with complex numbers well
try superposition if you have like you
know several waves superimposed or if
you have an integral a continuum of
superimposed waves which will see very
soon like fully eternal formulas on
written as formula extremely
inconvenient if you write them as cosine
transform big mess the kratom is full of
complex Exponential's is actually very
simple to do the math which is it is a
it is a very very time-saving tool
basically a lot of strange than that it
gives leads to some insights that the
real numbers do not give you that's an
additional convenience but having
Google's appreciated later for now it is
simply the mathematical convenience that
drives this mathematical wizardry yeah
yeah he's saying instead when the
angular frequency in the temporal
frequency are different or it's the same
because you initially defined Omega as
angular frequency i confuse those myself
usually the terms usually knew this only
know is called the frequency and it is
measured in Hertz a 2 pi nu is measured
in radians per second and this is
defined what about angular frequency
isn't it yeah
Oh as opposed to what else and guess I'd
understand the question
yeah what I mean is that that for
example take the propagating wave the
propagating wave is e to the I we say
forwarded KZ minus Omega T the standing
wave is cosine KZ e to the minus I Omega
T I'll go I'll go
I'll step ahead a bit and write
something awful
okay
this is actually a wave it is what we'll
see later it is called a diffraction
integral and if you can see it is
nothing even though it looks really
complicated it is just a superposition
right because the integral is a
summation and then what I have here is a
phaser and I put a lot a lot of these
values together what they all share is
is this pyramid to the I minus I Omega T
okay so I could actually to be strictly
correct there should really carry this
term I'm not really allowed to drop it
but because this is the same always I
might as well just as well not writing
so it basically saves me writing and
also confusion because as you know the
more things you write the more likely
you are to make mistakes so a typically
you know it is typical in these
derivations if you don't need something
you drop it right because if you start
carrying it around it it involves
mistakes so that's why I would drop them
so I can just as well say that if I have
a propagating wave I can just write it
as e to the ikj that's a propagating
wave have a standing wave I can just as
well write it cosine KZ okay they -
there - I Omega T is included right if I
have this diffraction integral I can
just write it without this term but
again the time is implicit if I have to
figure out what is the temporal
variation of the way I have to put it
back in and again I cannot do it if I
have two waves of different frequencies
if I have e to the I Omega let me write
- let me write another superposition get
to the I k1 Z minus Omega 1 T plus e to
the I a - Z minus Omega 2 T that's still
a valid superposition provided that the
ratios of these you know K 1 over Omega
1 equal see a 2 Omega 2 equal C still
valid superposition I'm not violating it
but I cannot do the phaser trick anymore
right I have to keep the
it was I am I get it terms or I will
make a mistake there's one more case
when if you produce any kind of
non-linearity we will say later and
again where we're going a little bit
ahead I'm in trouble
these are from the police okay we'll see
little bit later when we try to compute
the flux of electromagnetic energy the
something called the pointing vector
which is defined as the electric field
cross with the magnetic field okay if
you try to write the expression for the
pointing vector
you cannot just multiply the phasers the
electric and the magnetic field because
it's a product so you can use phasers
only when you have linear operations
such as a distance ray or integrals in
this case you can't we will see what you
do in this case you do a temporal
average but but there is a shoe if you a
cases where one has to be careful with
this phaser