Circular waveguides — meaning of the two subscripts in TE_mn / TM_mn mode notation What do the first and second subscripts represent in the standard mode labels for circular metallic waveguides?

Introduction / Context:
Modes in circular waveguides are designated TE_mn or TM_mn. Their field components satisfy Bessel-type radial dependence and sinusoidal azimuthal dependence. Correctly interpreting the subscripts is essential for determining cutoff frequencies, field distributions, and degeneracy among modes.

Given Data / Assumptions:

• First subscript m relates to angular (circumferential) variation.
• Second subscript n indexes radial eigenvalues (Bessel zeros or derivative zeros).
• Metallic, perfectly conducting circular boundary is assumed.

Concept / Approach:
In circular geometry, the transverse fields can be expanded in terms of Bessel functions J_m(k_r r) for the radial dependence and sin(mφ), cos(mφ) for the azimuthal dependence. The integer m counts the number of full sinusoidal variations around the circumference, while n orders the radial eigenvalue (the nth root that satisfies boundary conditions). Thus, the first subscript ties to the circumferential variation, and the second maps to the radial Bessel-function order (root number).

Step-by-Step Solution:

Verification / Alternative check:
Cutoff frequency tables list k_c a values corresponding to zeros of J_m or its derivative (for TM or TE respectively), indexed by n for each m, which aligns with this interpretation.

Why Other Options Are Wrong:

• A/C invert or mis-assign radial vs circumferential meanings.
• D: incorrect because a standard, accepted interpretation exists.

Common Pitfalls:
Confusing rectangular guide indices (m along a, n along b) with circular-guide indices (m azimuthal, n radial); forgetting derivative vs function zeros for TE vs TM.

Final Answer:
the Bessel function variations in radial direction and the number of full sine wave variations in circumference direction respectively