simplifying matters considerably.
We can just invert the derivative:
so first we find
The same sort of thing happens at each resonance. Since and X is always positive, vg < c. The same is not true for since the corresponding X factor can be positive or negative depending on whether is smaller or larger than a given .
From the definition of the mode numbers m and n, a TE00 mode
would have kx = ky = 0 - i.e. we'd have an ordinary plane wave
propagating straight down the waveguide at c.
Neither nor would depend on x or y
within the guide, only on z and t.
Both would be perpendicular to
like the sides of the guide, one would be parallel to
and the other would be parallel to .
Thinking in these terms, we can see several reasons why such a wave
is impossible. The most obvious is that
does not depend on x or y,
there is no way for it to change continuously to zero
inside the conductor, as it must. The same applies to .
However, we are not encouraged by Griffiths to visualize the wave as
a superposition of reflections (or, in this case, no
reflections) of a simple plane wave; in principle there might
be some exotic (x,y) dependence that would keep kx = ky = 0
and still satisfy MAXWELL'S EQUATIONS, so we go the formal route
and prove otherwise.
For any TE wave, Ez = 0 by definition. Equations (9.179) then become