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# analytical numerical dispersion - explanation of derivation

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#### confi999

##### Junior Member level 3
numerical dispersion

Hi,
In Taflove book - third edition on page 110, section 4.3 'Extension to three dimensions' we see how analytical numerical dispersion formula has been derived. A system of homogeneous equation was derived and then determinant of the coefficient matrix of that system was set to zero. This gave us the analytical numerical dispersion equation.

Can someone please explain this --
a) Why the determinant was considered zero and
b) Why it is required to get the dispersion formula. What is the connection.

In mathematics, if the determinant of a matrix is zero there is no inverse of that matrix. Is there any connection with this to the above.

You will see a similar procedure of dispersion equation derivation in the following paper-
https://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=920165

Please explain if you can. Many thanks.

Added after 3 hours 35 minutes:

Hi,

Further to the above, I have computed the determinant of the coefficient matrix ('coeffmat' here) for 3D FDTD using the 'Det[coeffmat] command iin Mathematica. This gave me determinant=0
Then how one gets the dispersion equation? Because the equation vanishes with that consideration of determinant equal to zero of the Taflove book (zero=zero). Am I missing something?
Thank you

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