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So we've looked at the dispersion relation for when
we used FDTD to solve the scalar wave equation.
Now let's look again at the dispersion relation but this time
for FDTD applied to solving Faraday's and Ampere's Laws.
And in order to look at this, the accuracy of FDTD, let's first examine the exact
dispersion relation so that we can use to compare to the numerical dispersion relation.
So this would be for EZ and HY.
We'll go ahead and do it for our one dimensional
problem that satisfy DHYDT 1 over mu DEZDX.
And then also DEZDT, 1 over epsilon DHYDX.
So we're going to follow a similar methodology
here as we did for the scalar wave equation.
We're going to write the form of our solution for EZ.
It's a function of X and T is equal to E knot. So we have some amplitude.
E to the J omega T minus KXX where K is the physical wave number. And HYX,
T is H knot our amplitude and E to the J omega T minus KXX.
And we're going to substitute these into the equations here. So into both of these.
So on the next slide here I'll expand this after we've applied them.
So on the left-hand side we had the partial time derivative.
So we get a J omega out in front. E to the J omega T minus KXX is 1 over mu.
And here we get a JKX out in front. Same exponential.
And then for this we're going to get omega mu
KX once we simplify minus E knot over H knot.
And for the second equation 1 over epsilon minus JKXH knot similar form and
when we simplify we get minus E knot over H knot is KX omega epsilon.
So therefore we get omega mu over KX is KX omega epsilon.
And then KX squared is omega squared over V squared.
And this is, of course, then where V is 1 over square root of mu epsilon.
So now we have our 1 D exact dispersion relation.