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So we've come up with the exact dispersion relation.
So now let's look at what the numerical dispersion relation is for our EZI and HYI plus a half and plus a half, which satisfies the 1-D
FDTD equations and I have typed these up our 1-D FDTD equations here so that they could be written nice and cleanly on the slide.
So we're going to go through a similar methodology as before on the next slide.
I'm going to let the form of our solution for EZIN is equal to E knot our amplitude.
E to the J omega N delta T minus -- KX and that has a tilde on top. I'm writing too high on the screen again. So this is the tilde.
So this is our numerical wave number. We're going to assume that our omega is equal to the physical angular frequency.
So I'm not going to put a tilde on that. And also the form of our solution for HY is also going to be the same except H knot is our amplitude.
Okay. And we're going to take these, substitute into the FDTD equations. All right.
So for the first one, once we do this, after a little bit of simplifying and canceling of common terms, we'll get this.
H knot and an E,E to the minus JK tilde X delta X over 2 and E to the J omega
delta T over 2 minus E to the minus J omega delta T over 2, all over delta T.
And on the right-hand side 1 over mu E knot, E to the minus JKX delta X, we cancel the common terms. So we just get minus 1 and delta X.
Now, I'm going to simplify this a little bit further. Move some stuff around.
Get H knot over E knot is equal to delta T over mu delta X minus 2 J sine KX delta X over 2 over 2 J sine omega delta T over 2
and we can simplify this one last time minus delta T over mu delta X sine KX delta X over 2 and sine omega delta T over 2.
So I've used an identity here to write our exponentials and cancel terms and now we're just left with the signs.
So that's then from the first equation. Once we do this for the second equation, we plug it in. You're going to get E knot,
E to the J omega delta T minus 1 over delta T over 1 over epsilon H knot E to the J omega delta T over 2. E to the minus JKX delta X over 2.
And hopefully I can fit this here. E to the JKX delta X over 2. All over delta X and our close parentheses there.
And then once we simplify a bit further, we're going to use our identity again.
We'll get minus epsilon delta X over 2, sine omega delta T over 2 over sine KX delta X over 2. So we've come up with two relationships.
This one from the second equation and here from the first equation.
So what we'll do is we'll set them equal to each other because they're both H knot over E knot.
So we can set them equal to each other and then we'll get our dispersion relation. This will be our 1 dimensional numerical dispersion relation.
So this is equal to sine omega delta T over 2 V delta T and squared. So this is our 1-D numerical dispersion relation.
And you can see again as we head for the scalar wave equation, this is not the same as the exact dispersion relation.