Rectangular waveguide – Which mode cannot propagate? For a standard hollow rectangular waveguide (perfectly conducting walls), identify the mode designation that is <em>not</em> supported as a propagating TE or TM mode.

Introduction:
Rectangular waveguides support a family of discrete transverse electric (TE) and transverse magnetic (TM) modes. The allowed indices depend on boundary conditions. Recognizing which index combinations are valid is essential for mode charts, cutoff calculations, and avoiding spurious propagation in microwave hardware.

Given Data / Assumptions:

• Lossless, air-filled rectangular waveguide.
• Mode notation TEmn or TMmn, where m and n are nonnegative integers describing half-wave variations along the wide (a) and narrow (b) dimensions.
• Dominant mode is TE10 for most rectangular guides.

Concept / Approach:

For TE modes, at least one index must be nonzero (m, n not both zero). That allows index combinations such as TE10, TE01, TE11, etc. For TM modes, both indices must be nonzero because a TM mode requires both transverse variations to satisfy boundary conditions with Ez ≠ 0 and Hz = 0. Therefore, TMm0 or TM0n do not exist as propagating modes.

Step-by-Step Solution:

Verification / Alternative check:

Standard tables list TEmn for m,n ≥ 0 with not both zero; TMmn for m,n ≥ 1—confirming that TM10 is disallowed.

Why Other Options Are Wrong:

TE15, TE12, TE01, and TM11 are all permissible modes (though their cutoffs differ).

Common Pitfalls:

Assuming TM modes are allowed with a zero index like TE; they are not.

Final Answer:

TM10.