In deriving the EM wave wave eqaution, we 
have to look at the governing laws of the 
electric field and the magnetic field
and how they play with each other. These are 
known as Maxwell's equations 
as you see me list them out here: 
You have Gauss' law, Faradays' law and 
Ampere's law. You may want to copy them 
down, if these are too messy for you
I am sure you can just goggle or wikipedia 
"Maxwell's equations" and you can see them 
in all their nice clean mathematical glory
So to interpret what these laws mean as a 
quick review for you guys, let's go through 
these one by one:
Gauss' law talks about how if you have a 
bubble in space
and this bubble encloses a certain charge 
and creates some kind of electric field
and when you add up all these electric fields, 
you can work out how much charge there is 
inside this bubble
so Gauss' law talks about how charges 
creates electric fields in space
For magnetic fields, however, you can't have a 
magnetic monopole inside, so you can only 
have 
magnetic field goes in from one side and 
leaves the other side and the sum always 
eqauls to 0
as you can never trap any magnetic 
monopole, so this is also saying that there is 
no magnetic monopole
Faraday's law is the one that talks about that 
how if you have a loop of wire
and you are changing the magnetic flux inside 
the loop, you can get some emf that goes 
around the loop
so how changing magnetic field gives 
changing electric field
and then you have Ampere's law, which the 
first parts talks about if you have a wire [with a 
current], 
you can create some kind of magnetic field 
around a loop
and then the second part comes about when 
you consider a capacitor being charged up
that while you have magnetic field here due to 
current, and magnetic field here due to 
current
but here, you also have magnetic field, but 
there is no current, because it's open plates. 
But the electric field is changing, so this is the 
last piece of the puzzle where you have 
changing electric field giving you changing 
magnetic field
so it is from here that you have your changing 
B gives changing E, and this is changing E 
giving changing B
and it is this symmetry that allows us to form 
the wave equation
so these are the basis from which will work 
and develop our wave equation
now to prove the wave equation and derive it 
in general, we will need some 
more advanced vector calculus, so we are 
going to go a little simpler and focus on
a particular type of solution to see if it 
conforms to our 4 Maxwell's equations
and that simpler solution is what's known as 
a plane wave
So when we say we are going to focus on our 
plane wave solution. The plane wave is more 
or less your 1D case in a 3D world
