in this module, we will be looking at solution
of helmholtz equation, okay.
.
we have defined what is helmholtz equation,
it is actually the one that does not contain
explicit time dependence, okay. so we want
to find out how to solve this helmholtz equation,
in generally turns out that it is actually
very difficult to solve this helmholtz equation.
but it is possible to pick certain, based
on some intuition, it is possible to pick
certain directions for the electric field,
okay.
and then see if our picked solutions or our
guessed solutions are actually consistent
or not, okay.
let us imagine that these waves, electromagnetic
waves, are being generated by some action
at sun, okay, which we can assume to be very,
very far away from us, right.
and let us also assume that these waves are,
of a particular frequency only.
although this may not be true, and it is not
true in fact in general.
but let us assume for simplicity that, these
waves which are generated on sun are of a
particular frequency.
so these waves are coming to us, okay. the
waves are actually spherical, okay. this is,
we are not going to prove that one here, we
can prove that one later.
but the waves which are generated are all
spherical, and then they would start expanding
like this.
you must have seen many movies, or you know,
you must have seen it if, when you drop a
pebble in the pond. that there would be this
expansion of spherical surfaces.
so this surfaces expand like this but by the
time they reach earth, the expansion would
be so large, that over the entire earth, or
at least over the region that i am considering,
let us say i put up my receiving antenna over
here to intercepts the sun's rays.
then over this antenna, which could be say
one metre by one metre, for example, that
wave which is coming although spherical becomes
locally like a plane wave, right.
see, it is like imagine a ball, okay. so you
imagine a ball and then imagine the radius
of the ball being increasing, increasing,
increasing.
and then you start to pick of a very small
region of the ball, okay.
and if you see that region that looks almost
like a plane of paper, rather than a curved
surface, it looks almost like a piece of plane
paper, right.
so this is the sphere that i have, okay, and
then if i now pick up a very small region,
which is what would happen, right.
see, this is the small region why because
this spherical surface which represents the
wave generated from sun, is actually quiet
large, right.
so, on that i am putting my small receiving
antenna, and seeing what is the variation
of electromagnetic field over here, on this
antenna.
so this is the small region, which i am seeing,
so although i would be looking at a curved
surface like this, right.
this is the curved surface that i am looking
at, but for all practical purposes, this can
actually be replaced by a plane graph, okay.
so this kind of waves are called as plane
waves, okay. and we will be looking at plane
waves, okay. and mathematically plane waves
are quite easy to specify, because they can
be specified easily, i will tell you in a
minute now.
but, physically plane waves cannot really
exist, okay, because mathematical plane waves
cannot exist, simply because, we will see
why they cannot exist.
but, the idea is that if a wave source is
sufficiently, sufficiently far away from the
place where you are looking for the waves,
then you can actually model that wave as a
plane wave, okay.
so, that is the whole idea of using plane
waves.
this is why sometimes, this is called as plane
wave approximation. and that is very widely
used in electromagnetic studies and optical
studies, okay. so, how do we define mathematical
plane waves.
mathematical plane waves can be defined by
writing, so mathematically plane waves can
be defined 
by associating e and h fields to be of this
form, which form, i will just in a minute,
i will draw that.
the form is this.
these are the electric field lines, okay.
these are the electric field lines of the
plane wave, okay. and these are the magnetic
field lines.
you can see that these electric and magnetic
field lines are all crossing at right angle.
so, you can see that, these are all crossing
at right angles, okay. so, this is like taking
up wired mesh and then putting it up in front
of you with one axis pointing to the electric
field.
this is the electric field, and this is the
magnetic field, e and h fields, okay. so,
these are called the plane waves, and this
plane waves have to be advancing, they have
to be propagating somewhere, right.
so, if you go back to the spherical wave,
you know, which is expanding like this, spherical
wave expanding like this, right.
it actually means that, i have these plane
waves, right, which are expanding all the
way, right.
so, these plane waves are all expanding all
the way over here, right.
and then, this is how the plane waves are
expanding.
so if i consider the direction in which these
plane waves are expanding as z axis, right,
okay.
let me consider the direction of expansion
of wave surface as z axis, then it is immediately
clear that electric field and magnetic field
both have to be in plane, that is perpendicular
to z axis, correct.
they have to be plane that is perpendicular
to z axis. so, for example, this is my axis
of propagation, the wave is propagating like
this, and this is my magnetic fields, okay,
or you can think of this board, that is there
at the back here, okay. as consisting of lines
which are electric and magnetic field lines,
okay. and then the board is advancing towards
you, the board is coming towards you in the
z direction.
so, this is the direction of wave propagation,
and these are the electric and magnetic fields,
okay. so, we are free to choose electric and
magnetic fields to be lying along x or y direction,
but they have to lie only in x and y direction,
and they have to be perpendicular, to each
other and they have to be perpendicular to
the plane that contains both, right.
.
so, e is perpendicular to h, which is perpendicular
to z axis, which is the direction of propagation.
this is the first rule for a plane wave, which
is propagating along z direction.
if the plane wave direction is considered
to be x axis, then e and h must be perpendicular
to each other and they have to be perpendicular
to x axis. in that case, electric and magnetic
fields will lie in yz plane, okay, it is as
simple as that.
so, you have electric fields and magnetic
fields.
now in xy plane and perpendicular to z axis.
now, within this xy plane, can the strength
of electric field vary.
we can rigorously show that the strength of
electric field and magnetic field cannot vary,
they have to remain constant.
this is a plane wave, the strength of electric
field and magnetic field remains constant
in xy plane or completely independent of xy
coordinates.
you can, kind of intuitively think about why
is this so. if they were to depend on x and
y, then there would be some curl, right, and
there would be some divergence.
but, if there is divergence, then del dot
e will not be equal to zero, right, and del
dot b is not equal to zero.
therefore, we can not have variation, okay.
we will not prove them rigorously, it is not
important.
but, take this point that, electric and magnetic
fields must be independent of x and y coordinates.
these conditions are sufficient for us to
consider electric field, magnetic field and
z axis, as a system of three mutually perpendicular
axis, right, so one electric field, magnetic
field and then you turn electric field and
magnetic field, it would propagate in the
direction of z axis.
now, you might rightfully ask, who stopped
you from considering propagation along minus
z direction, answer is no one.
actually, there could be exactly a wave which
is going along minus z direction also, because
the equations actually do not tell you that
you have to consider only forward propagating
equations.
they also tell you that you can consider,
in fact mathematically the solution would
be backward propagating, or negative z propagating
solutions as well, okay.
and, if you go to that electric field and
magnetic field should be in such a way that,
electric field to magnetic field will point
in the direction of z, then you have to appropriately
change the electric field and magnetic field
orientations, okay.
other than that, there is no problem if you
consider negatively propagating wave equation,
i mean wave solutions or waves.
but, of course, physically you might not have
a source, right.
if only there is a source on to the right
side there would be some waves which are going
in backward direction.
if the source are on the left side, and you
consider this as the right side, or the positive
z axis, the waves would only propagate to
the forward axis, i mean the forward z region,
okay. if there are some reflectors or scatterers
in between, then it is possible that, at any
given region of space there could be propagation
of both plus z and minus z waves.
and, those are the some examples that we will
see later, okay. so, to recapitulate the solution
for helmholtz equation is in general complicated.
to simplify those complications, we will assume
a plane wave approximation.
plane wave approximation is very nice, very
valid approximation, when the source of waves
are quite far away from the region, where
you are talking about this.
mathematically, plane waves can be defined
by associating electric and magnetic field
such that, they are both perpendicular to
the direction of propagation, say z axis,
okay. and, they remain constant in the xy
plane, or they remain independent of xy coordinates.
so, effectively what i have now is electric
field being consisting only of say, x component,
which is x hat ex, right.
and it should be independent of x and y.
so what can it be dependent on, only z, yes.
only when you keep moving far away from the
plane wave, then its amplitude might change,
amplitude would decrease, right.
otherwise, the amplitude is independent as
long as you are at a constant z plane, okay.
so, this is the expression for electric field
now.
similarly, magnetic field has to be along
y, hy of z. now let us see whether, whatever
we used some physical justification and intuition
actually, is also mathematically valid.
can we show that, these assumed solutions
are actually solutions, if they are what is
the nature of this ex of z and hy of z, okay.
.
so, today that let us recall helmholtz equation,
that is del square plus omega square by v
square into e is equal to zero, and instead
of writing omega square by v square every
time, let me introduce an expression, okay,
which a constant k, which is equal to, in
such that k square is equal to omega square
by v square.
so let me introduce that, so that this equation
becomes del square plus k square, okay, into
e is equal to zero.
now, expand this equation, what is del square,
del square is del square del x square del
square by del y square plus del square by
del z square, electric field e, electric field
is e is nothing but x hat ex of z, right,
plus k square, that is x hat ex of z is equal
to zero, correct.
now, ex is only function of z, it is not a
function of x or y, so there is no point writing
this del square by del x square and del square
by del y square terms.
they all cancel each other and then there
is x hat, x hat everywhere, so which simply
means that i can drop the vector also from
this condition and then just replace this
with the scalar equation. and i get del square
ex by del z square plus k square ex is equal
to zero, right.
and since, ex is only function of z, there
is no need for me to write down this as del
square by del z square.
i can simply write this as d square ex by
dz square, okay. this is the equation that
is highly simplified from helmholtz equation,
okay, based on our ideas that we have discussed
previously.
and we want to see what is the solution for
electric field ex.
how do we solve this equation, well i do hope
you remember your solutions for differential
equations.
so the solution that we were looking for was
for this equation d square ex by dz square
plus k square ex equal to zero.
.
the solution of that equation is, have d square
ex by dz square is equal to minus k square
ex, correct.
the solution for this would be ex of z is
equal to some constant ae to the power jkz
plus some constant be to the power minus jkz.
does it actually get satisfy this equation,
can you just verify that this is actually
the equation.
i leave this as an exercise to you, okay.
this is a small exercise to you to show that
this assumed solution, or this guessed solution
in deed satisfies original equation, okay,
which is this one, will call this as some
equation one, okay, equation one.
you should verify that, for example, you turn
of b, okay, then what would happen, we have
ae power jkz, so differentiating one with
respect to z you will get jk, twice you will
get jk square, jk square is minus k square.
so minus k square into ex is equal to minus
k square into ex.
therefore this is the solution.
similarly, minus jk would also be the solution.
so it is up to you now, to whether retain
a plus kz solution or a minus kz solution,
mathematically both exist.
but if you now say that, well you know i do
not have any wave, any physical source which
would actually push the waves along minus
z direction, then i can make this equal to
zero, okay.
if not then, i do not have to make it equal
to zero.
so this is called as the forward wave, and
this is called as the backward wave.
i am using forward and backward in energy
with transmission line, which we are going
to study after the wave modules, okay. based
on that, any wave which is going along positive
z direction is forward wave for me, any wave
which is going along minus z direction is
called backward for me, okay.
so that is my simple, short notation that
i am going to use, okay.
i have ex of z is equal to considering only
the forward wave solution i have ex of z is
equal to a, which is some constant and instead
of talking about a which combines no intuition,
let us call this as e0, okay, e0 stands for
some amplitude of electric field.
so, you have e0e power jkz as the solution.
do you recognize what has actually happened
over here.
the expression for electric field is depending
on z, correct.
but it is actually a complex number, right,
if you try to sketch this one, you will not
able to sketch this.
it would actually be a complex number, you
will have to sketch the real part first, sketch
the imaginary part first, right, or you can
interpret this as a phasor, right.
so, if you actually think of this in terms
of the phasor with real and imaginary axis,
then corresponds to a phasor, which is rotating,
with an angular velocity of kz.
so, you fix z and fix k, then it would point
to a particular direction, but if you keep
moving along z, then this would actually be
a phasor, which is rotating, okay. we also
satisfied that its amplitude is changing only
with z, yes because there is no change in
x and y direction, right.
so this is just a wave which is propagating
along z direction, right.
the second thing that you have to notice is
the, this is a phasor.
therefore then, it is not the complete time
dependent form.
so, if you want to find the time dependent
form, you have to multiply this ex of z by
e power j omega t, then take the real part
of it, right.
so, to obtain the time dependent form, which
is ex of zt, and what do you get if you do
this operation, multiply by e power j omega
t.
you get e power j omega t into, e power j
omega t plus kz, okay.
and with that if you go back here, you have
electric field here as real part of ex of
z, and substitute for ex of z here i get e0
e power j omega t minus kz, and real part
of this is nothing but e0 is real.
therefore e0 comes out of the real operation,
we have e0 cos of omega t minus kz. this is
the expression for electric field, okay. this
is in the scalar form.
if you want to write down the expression of
the electric field in the vector form, you
can attach the appropriate vector direction,
right.
this electric field would always be directed
along the x axis. and if you want to fix z
and t and everything, this cos term is fixed,
then it would point with an amplitude of e0
time something, whatever the cos value is,
something and it would always point along
x, okay.
there are two things we can do now, fix t
and see what happens with respect to z, and
fix z and see what happens with respect to
time.
this is like, i take an oscilloscope i put
it up in to air, okay. and this oscilloscope
display should tell you the electric field,
okay, that of the form cos omega t minus kz.
and i do not, i am satisfied with only one
oscilloscope at this particular point insert,
have two, three different oscilloscopes, okay,
kept at two different or three different points
along z direction, okay.
now, if i were to say, this is z equal to
zero and this is z equal to something, this
is z equal to something, rather than talking
about z, it is easier for me to talk in terms
of kz, okay. so, kz equal to zero, kz equals
pi by two, kz equals pi. i have got three
different oscilloscopes at values of kz equals.
zero, kz equals pi by two, and kz equals pi.
this kz equal to zero is my reference, okay.
now, with that, if i were to see what is the
display on the oscilloscope, right.
what would be the display that i would see.
.
go back to the expression, take kz is equal
to zero with kz equal t zero the expression
for electric field will be x hat e0 cos of
omega t.
so if you were to look at the oscilloscope
display, you would actually see, a display
that would look like this, correct.
this is the cosine wave form, this is the
time.
this is the axis with respect to time, so
this is the time t equal to zero, with an
amplitude of e0, right.
this will have an amplitude of e0, and it
would go as in this particular fashion, okay.
now, if you try what happens at kz is equal
to pi by two, get x hat e0 cos of omega t
minus pi by two.
now, cos of omega t minus pi by two is sin
omega t, right.
so this would be cos omega t minus, so it
would be cos omega t, cos pi by two, which
is zero, minus, sorry, plus sin omega t sin
pi by two.
so this would actually be equal to x hat e0
sin omega t, right.
so you look at your oscilloscope display,
the oscilloscope display would now show here,
right, a sin wave, at time t equal to zero
with a maximum amplitude of e0 again.
but, if you notice at kz equal to zero and
kz equal to pi by two, you will actually see
a pi by two phase difference, right.
this wave at kz equal to pi by two is actually
lagging the first wave by a value of pi by
two, right.
similarly, now when you write kz is equal
to pi, what you get is, x hat e0 cos of omega
t minus pi, right.
and this is cos omega t cos pi, sin will be
anyway will be zero.
so this will be minus x hat e0 cos omega t.
so if you look at the oscilloscope display,
the oscilloscope display will show you a wave
that would look like this, with an amplitude,
initially at t is to zero, as minus e0, right.
so, you can actually see that if you were
to put a point of reference, you know some
small object over here, right.
this small object, as you see is at different
times, appearing at different times, right.
you do not have to do this one with respect
to time.
you could actually fix time, okay.
