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Let's consider now the accuracy that we can expect from an FDTD solution to the one dimensional scalar wave equation.
In order to consider the numerical accuracy let's first look at what the exact dispersion relation is for our unknown U.
So the exact dispersion relation for our unknown U, function of X and T, that satisfies our 1 D scalar wave equation.
And, of course, you may possibly remember this from an earlier class.
Electro magnetics course, introductory electro magnetics course, but let's go ahead and look at it to remind ourselves.
So let us consider -- whoops -- let's consider the wave solution a continuous sinusoidal wave solution of the form E to the J -- whoops.
Let me delete that. So E to the J omega T minus KX.
Where K is the physical wave number where it can be complex and general but here we're going to assume it is real, because sigma is equal to 0.
We'll do this for free space. What we want to do here is we want to substitute this solution, the form of our solution into the wave equation.
So substitute into this into our scalar wave equation, and that's going to give us
J omega squared once we do the second partial derivative with respect to time.
And E to the J omega T minus KX, and we'll get a V squared out in front for the spatial, second spatial derivative, minus JK squared.
E to the J omega T minus KX. And this will give us an omega is equal to plus or minus VK, where V is defined as 1 over square root of mu epsilon.
So this is our exact dispersion relation. Now let's look at what it is for when we solve this using FDTD.