Electromagnetic wave patterns in waveguides for different modes like TE10, TM11

Question

I have read about different waveguide and their different modes like TE10, TE11, etc. In many books, the EM wave patterns are given, where they were said to be drawn from the waveguide's equations. But I can't understand how to draw this pattern.
I have read this link: http://www.rfcafe.com/references/electrical/waveguide.htm. Here TE10, and TM11 modes' EM wave patterns are given.
Can anyone tell me how they draw this pattern? I know it is difficult to explain it in words, but can you share any link or lecture that will let me understand it better?
This is not homework question. It is my curiosity to understand it better. Thanks.

hyportnex · Answer

Denote $$K_{mn}=frac{sqrt{epsilon_m epsilon_n}}{sqrt{m^2frac{b}{a}+n^2frac{a}{b}}}$$ and $$epsilon_0=1 text{ and }epsilon_m=2 text{ if }m>1$$

The $TE_{mn}$ modes have the $E$-field components 
$$ begin{matrix}
E_x = K_{mn} frac{n}{b} cosleft(frac{mpi}{a}xright) sinleft(frac{npi}{b}xright) 
E_y = -K_{mn}frac{m}{a} sinleft(frac{mpi}{a}xright) cosleft(frac{npi}{b}xright) 
E_z = 0 
end{matrix}$$
This vector field $left(E_x,E_yright)$is plotted in any $z$ cross-section. There are similar equations for the $H$ field but that also has an $H_z$ longitudinal component.

Electromagnetic wave patterns in waveguides for different modes like TE10, TM11

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