Finding Ratio of Lowest Cutoff Frequencies for Waveguide - Quick Question

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Homework Statement


I have to find the ratio of the lowest TM cutoff frequency to the lowest TE cutoff frequency.


Homework Equations





The Attempt at a Solution


I have derived the general form of the cutoff frequency (which seems to be the same for both). I get the right answer if I take TE10 and TM11; but why is this? I know why you can't have TE00 or TM00, but why can't TM take TM10?

Thanks
 
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What shape waveguide is this in reference to?

Assuming you are talking about a rectangular waveguide, what happens to the component of the electric field along the axis of the waveguide for TM10 and TM01? What does that make all the other components? Are those really TM modes?
 
To solve this, I first used the units to work out that a= m* a/m, i.e. t=z/λ. This would allow you to determine the time duration within an interval section by section and then add this to the previous ones to obtain the age of the respective layer. However, this would require a constant thickness per year for each interval. However, since this is most likely not the case, my next consideration was that the age must be the integral of a 1/λ(z) function, which I cannot model.
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