So let us now analyze the solution from the previous lecture in more detail, what we will establish
here is the dispersion of relation.
We will be able to relate these quantities, Omega and Kate, that I've introduced into previous lecture.
And also in this lecture, we will talk briefly about WAF packet's.
So the solution from the previous lecture was that the electric field and the magnetic field are both
electromagnetic waves.
So these are exponential functions with imaginary arguments and we have a fluctuation in time t and
in the coordinate R and these equations where the solution of this differential equation here.
So this is the wave equation for E and the wave equation for P.
So what we must do to determine the relation of Omega and K, we must just take this equation, the
solution here and put it into the equation.
So what I've done is I've just given you the solution and basically we made an ANZAAS that solves this
equation.
So now let's check if it is really a proper solution.
And also we will be able to determine these coefficients.
So let's take this equation for E!
And put it into this wave equation for E, what we will get is from this term we will get one over C
Square.
And then we have here the second derivative with respect to time, this gives us a square and omega's
square.
And then, of course, E zero times E to the I and so on.
So this is this term here multiplied by this.
And also we must of course, calculate here it is not class operator.
So once again, it's maybe more easily to go to one dimension where this lipless operator just becomes
the second derivative of respect to, let's say, X, and then we have here K times X.
So what we will get as the the river Tife will get minus I times K Square.
So this is this one here and then of course the whole term here, which stays because when you differentiate
in the exponential function it will just stay the exponential function.
So this is the what we get when we take this term and put it into the wave equation.
And we know that this must be equal to zero.
So we can just divide by this term here.
And we now know that this bracket here must be equal to zero.
Most of what you will see is you can also take the solution for B, put it in here and you will get
a very similar relation where you can divide by zero times as exponential function.
So you will get the very same relation we get.
In both cases that I Omega's square of a C Square is equal to this term here, minus K Square.
So of course we know what I square is.
It is minus one.
So let's simplify all of these minus signs here so we get minus a square of a C square and here we get
a plus sign because here we have a minus and then we get another minus from the square.
So we know that Omega is equal to plus or minus C times K, so Y plus or minus because we have this
square here.
So this is the dispersion relation of the electromagnetic wave.
And as it turns out, we know that C is the velocity of light and we can interpret K as a wave vector.
So this is like the two pi over the wavelength of the electromagnetic wave and we have Omega, which
is essentially two pi over the period of this fluctuation of the electromagnetic wave.
So we have the frequency that is linearly related to the wave factor and this proportionality constant
here is the velocity of light.
And also we know that we get a positive and a negative sign here.
So it means that also we can write it in a different way.
We can write down that K is equal to plus minus Omega oversea.
So it means that the key vector can be oriented along two different directions, you could say.
So it means the wave can propagate forwards and backwards.
And in fact, this relation here is not restricted to only fixed.
Constant is C, but you can just scale Omega and you can scale K.
So when you take some, for example, light from the sun and you analyse it, you will see that for
every portion of this light.
So to see you, this relation is true.
Omega is always linearly related to the way Vector K, but the light consists of many, many waves that
all have different frequencies and have different weigh factors.
These are the different colors of light.
So for example, red light has different by a factor than blue light.
And also they have different frequencies.
But of course they always have the same velocity.
But what this means is that the most general solution of the wave equation is not just a single wave,
as I've written down here, but it's superposition of many waves.
So let's take these waves here, which are, for example, for electric field zero times this exponential
function and and add them up for different characters.
And this is what I have done here, so adding up different characters since K is a is not a discrete
quantity, but a continuous quantity.
It's not a sum, but it's an integral part.
We well, we do not add up different waves, but where we integrate over these waves with respect to
K and then we have here another function which tells us by which amount the different waves with a different
way factors contribute to our total wave.
So in general, the light is a wave packet's that is the superposition of many different individual
waves that are characterized by one particular K vector and by one particular frequency Omega.
And this distribution function or test is a weight function, F of K can be considered the full year
transform of this function here.
So what is what this means is that when you know the different contributions in the way of packages,
for example, you know that when you analyze light from the sun and tells you it consists to five percent
out of light with a way effect or K one and five percent out of line with the character K two and so
on, then you know exactly what your total wave looks like.
So the information about this distribution function is sufficient for knowing the whole function because
it is just a different representation.
It is the four year transform of this function here.
Now we know about this person relation of light and also about WAF packet's, so please don't worry
if this last part was a bit too theoretical for you in the next lecture, things will become more simple
again.
And I want to show you what an electromagnetic wave looks like and how the wave vector K is oriented
with respect to E and also to be.